Fundamentals of Advanced Mathematics explains the basic fundamentals of mathematics by introducing it to the readers. It
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Table of contents :
Cover
Title Page
Copyright
ABOUT THE AUTHOR
TABLE OF CONTENTS
List of Figures
List of Tables
List of Abbreviations
Preface
Chapter 1 Fundamentals of Mathematics
1.1. Introduction
1.2. Proof And Mathematical Argument
1.3. Sets, Relations, And Functions
1.4. Construction And Properties Of Number Systems
1.5. Some Number Theory
1.6. Case Study: Analysis On Mathematics Fundamental Knowledge For Mathematics Engineering Courses Based on A Comparative Study Of Students’ Entry Performance
References
Chapter 2 Algebra And Basic Math
2.1. Fundamentals Of Algebra
2.2. Operations On Monomials and Polynomials
2.3. Linear Equations In One Variable
2.4. Problems To Solve
References
Chapter 3 Number Theory And Number System
3.1. Introduction
3.2. Number Theory
3.3. Facts About Number Theory
3.4. Number Systems, Base Conversions, And Computer Data Representation
3.5. Conversions
3.6. Number Systems
3.7. Conclusion
References
Chapter 4 Relations And Functions
4.1. Introduction
4.2. Binary Relations
4.3. Functions
4.4. Case Study: A MathematicalAlgorithmic Approach To Sets
References
Chapter 5 Propositional Logic
5.1. Propositional Logic
5.2. Introduction
5.3. Connectives
5.4. Tautologies
5.5. Contradictions
5.6. Contingency
5.7. Propositional Equivalences
5.8. Inverse, Converse, And ContraPositive
5.9. Duality Principle
5.10. Predicate Logic
5.11. WellFormed Formula
5.12. Quantifiers
5.13. Nested Quantifiers
5.14. Rules Of Inference
5.15. Operators And Postulates
5.16. Semigroup
5.17. Group
5.18. Abelian Group
5.19. Cyclic Group And Subgroup
5.20. Partially Ordered Set (Poset)
5.21. Hasse Diagram
5.22. Linearly Ordered Set
5.23. Lattice
5.24. Case Study 1: The Pragmatics Of Telling The Truth
References
Chapter 6 Graph Theory
6.1. Introduction
6.2. Links And Their Structures
6.3. Basic Structural Properties
6.4. Graph Theory Trees
6.5. Graph Theory Application
6.6. A Graph—Theoretic Data Model For Genome Mapping Databases
6.7. Case Study: Applying Graph Theory To Interaction Design
References
Chapter 7 Mathematical Induction And Recursion
7.1. Introduction
7.2. The Principle Of Mathematical Induction
7.3. Proof By Induction: Introduction
7.4. Induction And Recursion
7.5. Strong Induction
7.6. Case Study: The Flipping Glasses Puzzle
References
Chapter 8 Cardinality
8.1. Introduction
8.2. What Is Cardinality
8.3. Types Of Cardinality
8.4. Types Of Sets
8.5. Subsets
8.6. Sets With The Same Cardinality
8.7. Set Theory Symbols
8.8. Boolean Algebra
8.9. Values Of Cardinality
8.10. Elementary Theorems
8.11. Advanced Theorems
8.12. Cardinality Of The Continuum
8.13. Controversies
8.14. Conclusion
References
Index
Back Cover
FUNDAMENTALS OF ADVANCED MATHEMATICS
FUNDAMENTALS OF ADVANCED MATHEMATICS
Alberto D. Yazon
ARCLER
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www.arclerpress.com
Fundamentals of Advanced Mathematics Alberto D. Yazon
Arcler Press 2010 Winston Park Drive, 2nd Floor Oakville, ON L6H 5R7 Canada www.arclerpress.com Tel: 0012892917705 0019056162116 Fax: 0012892917601 Email: [email protected] ebook Edition 2020 ISBN: 9781774073919 (ebook) This book contains information obtained from highly regarded resources. Reprinted material sources are indicated and copyright remains with the original owners. Copyright for images and other graphics remains with the original owners as indicated. A Wide variety of references are listed. Reasonable efforts have been made to publish reliable data. Authors or Editors or Publishers are not responsible for the accuracy of the information in the published chapters or consequences of their use. The publisher assumes no responsibility for any damage or grievance to the persons or property arising out of the use of any materials, instructions, methods or thoughts in the book. The authors or editors and the publisher have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission has not been obtained. If any copyright holder has not been acknowledged, please write to us so we may rectify.
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ABOUT THE AUTHOR
Dr. Alberto Dolor Yazon is an Associate Professor and Chairperson of Curriculum and Instruction Development and Quality Assurance at the Laguna State Polytechnic University, Philippines. He obtained his Doctor of Philosophy in Mathematics Education at Philippine Normal University, Manila. He authored the published books Learning Guide in Methods of Research, Assessment in Student Learning, and Fundamentals of Advanced Mathematics. His research paper presentation, publication and areas of interest include problem solving, mathematics proficiency, motivation, coping mechanism, selfesteem, selfefficacy, adversity quotient, and assessment of graduate’s competencies. He has been a resource person for different topics related to Mathematics, Assessment of Learning, Statistics, and Research. He seats as a panel of examiner and adviser in master’s thesis and doctoral dissertation among graduate students. He is also an Accreditor in the Accrediting Agency of Chartered Colleges and Universities in the Philippines, Inc. (AACCUP) and had accredited five prestigious State Universities and Colleges (SUCs) in the country. His life’s principle is “Passion, commitment, humility, and quality are significant predictors of one’s destiny.”
TABLE OF CONTENTS
List of Figures ........................................................................................................xi List of Tables .......................................................................................................xiii List of Abbreviations ............................................................................................xv Preface........................................................................ ......................................xvii Chapter 1
Fundamentals of Mathematics ................................................................... 1 1.1. Introduction ........................................................................................ 2 1.2. Proof And Mathematical Argument ..................................................... 5 1.3. Sets, Relations, And Functions .......................................................... 10 1.4. Construction And Properties Of Number Systems ............................. 21 1.5. Some Number Theory ....................................................................... 27 1.6. Case Study: Analysis On Mathematics Fundamental Knowledge For Mathematics Engineering Courses Based on A Comparative Study Of Students’ Entry Performance................ 30 References ............................................................................................... 35
Chapter 2
Algebra And Basic Math .......................................................................... 37 2.1. Fundamentals Of Algebra ................................................................. 38 2.2. Operations On Monomials and Polynomials..................................... 40 2.3. Linear Equations In One Variable ...................................................... 46 2.4. Problems To Solve............................................................................. 58 References ............................................................................................... 64
Chapter 3
Number Theory And Number System ..................................................... 65 3.1. Introduction ...................................................................................... 66 3.2. Number Theory................................................................................. 68 3.3. Facts About Number Theory .............................................................. 70 3.4. Number Systems, Base Conversions, And Computer Data Representation ....................................................................... 71
3.5. Conversions ...................................................................................... 78 3.6. Number Systems ............................................................................... 80 3.7. Conclusion ....................................................................................... 84 References ............................................................................................... 85 Chapter 4
Relations And Functions .......................................................................... 87 4.1. Introduction ...................................................................................... 88 4.2. Binary Relations................................................................................ 94 4.3. Functions .......................................................................................... 98 4.4. Case Study: A MathematicalAlgorithmic Approach To Sets............. 110 References ............................................................................................. 114
Chapter 5
Propositional Logic................................................................................ 115 5.1. Propositional Logic ......................................................................... 116 5.2. Introduction .................................................................................... 117 5.3. Connectives .................................................................................... 120 5.4. Tautologies...................................................................................... 122 5.5. Contradictions ................................................................................ 123 5.6. Contingency ................................................................................... 123 5.7. Propositional Equivalences ............................................................. 124 5.8. Inverse, Converse, And ContraPositive ........................................... 125 5.9. Duality Principle............................................................................. 126 5.10. Predicate Logic ............................................................................. 127 5.11. WellFormed Formula ................................................................... 127 5.12. Quantifiers.................................................................................... 127 5.13. Nested Quantifiers ........................................................................ 128 5.14. Rules Of Inference ........................................................................ 128 5.15. Operators And Postulates .............................................................. 132 5.16. Semigroup .................................................................................... 135 5.17. Group ........................................................................................... 136 5.18. Abelian Group .............................................................................. 137 5.19. Cyclic Group And Subgroup ......................................................... 137 5.20. Partially Ordered Set (Poset) .......................................................... 138 5.21. Hasse Diagram ............................................................................. 139 5.22. Linearly Ordered Set ..................................................................... 140
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5.23. Lattice........................................................................................... 140 5.24. Case Study 1: The Pragmatics Of Telling The Truth ......................... 142 References ............................................................................................. 146 Chapter 6
Graph Theory ........................................................................................ 147 6.1. Introduction .................................................................................... 148 6.2. Links And Their Structures ............................................................... 152 6.3. Basic Structural Properties .............................................................. 153 6.4. Graph Theory Trees ......................................................................... 155 6.5. Graph Theory Application ............................................................... 159 6.6. A Graph—Theoretic Data Model For Genome Mapping Databases ....................................................... 171 6.7. Case Study: Applying Graph Theory To Interaction Design .............. 175 References ............................................................................................. 180
Chapter 7
Mathematical Induction And Recursion ................................................ 181 7.1. Introduction .................................................................................... 182 7.2. The Principle Of Mathematical Induction........................................ 182 7.3. Proof By Induction: Introduction ..................................................... 184 7.4. Induction And Recursion ................................................................ 195 7.5. Strong Induction ............................................................................. 199 7.6. Case Study: The Flipping Glasses Puzzle ......................................... 203 References ............................................................................................. 209
Chapter 8
Cardinality ............................................................................................ 211 8.1. Introduction .................................................................................... 212 8.2. What Is Cardinality ......................................................................... 212 8.3. Types Of Cardinality ....................................................................... 213 8.4. Types Of Sets .................................................................................. 215 8.5. Subsets ........................................................................................... 222 8.6. Sets With The Same Cardinality....................................................... 223 8.7. Set Theory Symbols ......................................................................... 224 8.8. Boolean Algebra ............................................................................. 226 8.9. Values Of Cardinality ...................................................................... 227 8.10. Elementary Theorems .................................................................... 228 8.11. Advanced Theorems ...................................................................... 229
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8.12. Cardinality Of The Continuum ...................................................... 230 8.13. Controversies ................................................................................ 230 8.14. Conclusion ................................................................................... 233 References ............................................................................................. 235 Index ..................................................................................................... 237
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LIST OF FIGURES Figure 1.1. Graph of a continued fraction Figure 1.2. Pretest result based on students’ entry performance from academic’s session 2008/2009 to 2009/2010 Figure 1.3. Pretest result for male students from academic’s session 2008/2009 to 2009/2010 Figure 4.1. Cartesian product Figure 4.2. Relations and functions Figure 4.3. Onetoone function Figure 4.4. Onto function Figure 4.5. Bijections Figure 4.6. Inverse function Figure 5.1. Hasse diagram Figure 5.2. Lattice Figure 5.3. An example of lattice Figure 5.4. Not a lattice Figure 6.1. Graphical representation of real network Figure 6.2. Basic graph representation of transport network Figure 6.3. Planar and nonplanar graphs Figure 6.4. Simple and multigraph Figure 6.5. Tree image Figure 6.6. Forest tree Figure 6.7. G is a connected graph and H is a subgraph of G. Figure 6.8. Circuit rank Figure 6.9. Image depicting Kirchhoff’s theorem illustration Figure 6.10. Spanning tree illustration Figure 6.11. Types of elements of the bipartite graph G Figure 6.12. The graph G for n = 2 and q = 3 xi
Figure 6.13. The DNA double helix and SNP assembly problem Figure 6.14. The input for the vertex cover algorithm Figure 6.15. The set {2, 4, 5} is a minimum vertex cover in this computer network Figure 6.16. Dual graph of the map of India Figure 6.17. The cells of a GSM mobile phone network Figure 6.18. A reentrant knight’s tour on the 8×8 chessboard Figure 7.1. Visualization of blocks and numbers Figure 7.2. Boxy triangle for visualization Figure 7.3. (a) A triangle: the first step for calculating the area, (b) vertical line drawn as the second step. Figure 7.4. From box triangle to a larger box Figure 7.5. After pulling off the last column, the shape of the large triangle box Figure 7.6. Rectangle shaped box when added extra boxes to the triangle Figure 7.7. Adding boxed with the same width of the triangle and also the height Figure 7.8. Pictorial depiction of candy bar Figure 7.9. Breakdown of candy bar for strong induction Figure 7.10. Glasses as discussed in puzzle Figure 7.11. First and last glasses inverted Figure 7.12. Second and fourth glasses inverted Figure 7.13. First and third glass inverted Figure 8.1. A∪B Figure 8.2. A∩B
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LIST OF TABLES Table 5.1. OR connectives Table 5.2. Truth table for AND connection Table 5.3. Truth table for negation connection Table 5.4. Truth table for implication/ifthen (→) connection Table 5.5. Truth table for if and only if (⇔) connection Table 5.6. Truth table of tautologies Table 5.7. Truth table of contradictions Table 5.8. Truth table of contingency Table 5.9. Testing by 1st method (Matching truth table) Table 5.10. Testing by 2nd method (Biconditionality) Table 5.11. Table of rules of inference Table 8.1. Table of set theory symbols
LIST OF ABBREVIATIONS
FKAB
Faculty of Engineering and Built Environment
GSM
Groupe Spécial mobile
MOE
Ministry of Education
POSET
partially ordered set
RAD
rapid application development
SNP
single nucleotide polymorphism
UIMS
user interface management systems
PREFACE
Fundamentals of Advanced Mathematics has been written with the objective of presenting readers with the foundation of some of the fundamental concepts of advanced methods. The study material provided in this book will contribute to widen the understanding of mathematical topics that further help in physics and several other concepts of engineering. The concepts discussed in this book are advanced since they are developed on the assumption that the reader has already understood the most important basic parts of a Mathematics degree. There are certain basic calculations, comprising no more than rudimentary arithmetic and the very basic laws of chance. After an introduction of basic number theory, number system linear algebra and basic math, this study will offer a deep understanding of advanced concepts such as relations, induction, etc. Increased utilization of the advanced formulas and equations is a parameter that always promises quicker results to mathematical problems. Achieving this aim is best helped by a detailed look through a scientific approach to solve the problems. To tap into everyone’s creativity for achieving a common goal can be a daunting task and that’s precisely what has given room for the development of analyzing the different methods into a complete field in itself. There is a plethora of information out there on the fundamental topics of advanced mathematics. The existing studies can at best give clues on the basics of advanced methods, but it is the thorough study of individual concepts that can give rise to more advanced methods that can work in real life. The subject matter of this book starts from establishing a clear picture of the fundamentals of mathematics. The education of this approach will contribute to widen the understanding on improving knowledge of basic methods. This would be supported by reallife case studies at the end of each chapter to enable the reader for achieving direct results. Next focus would be on number system and number theory. Subsequently, relations and functions together with propositional logic and cardinality. Towards the end, a comprehensive detail of mathematical induction and graph theory is discussed.
Above is a very simple anecdote of the utilization of fundamentals of advanced mathematics and a complete study has much more to offer. I look forward to the reader for achieving valuebased results by using the methodologies prescribed in the book. The constructive criticism and the feedback would be most welcome.
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1 Fundamentals of Mathematics
CONTENTS 1.1. Introduction ........................................................................................ 2 1.2. Proof And Mathematical Argument ..................................................... 5 1.3. Sets, Relations, And Functions .......................................................... 10 1.4. Construction And Properties Of Number Systems ............................. 21 1.5. Some Number Theory ....................................................................... 27 1.6. Case Study: Analysis On Mathematics Fundamental Knowledge For Mathematics Engineering Courses Based on A Comparative Study Of Students’ Entry Performance................ 30 References ............................................................................................... 35
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The learning of mathematics in schools represents a basic preparation for adult life and a gateway to a vast array of career choices. In a social perspective, proficiency in mathematics is essential for functioning in everyday life, as well as for success in increasingly technologybased workplace. Students who take higherlevel mathematics and science courses which require strong fundamental skills in mathematics are more likely to attend and to complete college. The globalization of markets, the spread of information technologies, and the premium paid for workforce skills all emphasize the growing need for proficiency in mathematics. There is a widespread interest among industrialized countries in improving the levels of mathematics achievement in schools. Apart from the economic benefits, it is argued that this would bring about economic progress by better preparing young people for the numeracy demands of modern workplaces, and raising the overall skill levels of the workforce. Also, there are social benefits tied to improving access for larger numbers of young people to postschool education and training opportunities and laying stronger foundations to skills for lifelong learning. The importance of mathematics extends beyond the academic domain. Young people who transition to adulthood with limited mathematics skills are likely to find it difficult to function in society. Basic arithmetic skills are required for everyday computations, and sometimes for job applications. Additionally, competency in mathematics skills is related to higher levels of employability. Ever since the influence of high school students’ mathematics skills on later earnings has grown steadily. With this overgrowing importance and demands of mathematics, the introductory part of this chapter includes the basic concepts of mathematics which are needed to advance to the next higher level of learning the subject. This chapter sets the context for the book – it briefly explains what basics of mathematics mean – the target audience and the various concepts related to numbers. Multiplying and dividing large numbers, simplifying fractions and converting percentages, handling square roots and exponents – these are few of the important skills that form a solid foundation on which all the concepts of mathematics are built.
1.1. INTRODUCTION For any student looking forward to the course of advanced mathematics, it is important to understand the key objectives behind the desire to take this
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course. The chapter begins with understanding the reasons for undertaking this course – whether they are career driven, aimed at gaining admission into formal education, or as a hobby or passion. It doesn’t matter if the person belongs to the category of the student of the high school, interested in preparing for various difficulty levels associated with mathematics, or the complexities of the challenges related to the classes of the higher maths, or an adult who possess the requirement or the need for a kind of the refresher within the maths so that they would be able to do the preparation for their novel career, or the one who only desires or wants to maintain or keep their minds quite active as well as sharp to be able to function properly. Thus, one would not be denying the fact that possession of the great level of the maths or the arithmetic as well as the prealgebra is of utter importance and is also a necessity in the world everyone is living at present. Having a sound or the proper knowledge of the various fundamentals of the mathematics would be helping in the following: •
Having an increment in the chances or the opportunities related to the success whether it be the high school or the class of the math in the college as well. • Provides with the aid to prepare for a great career in the field which actually possess the requirement or the need an active and a strong basic as well as the foundation within the maths having an inclusion of the various other subjects such as: the economics, building up of the various trades, the medicine as well as the engineering. • Provides with the support to help in strengthening the skills of having a critical thinking as well as the analytical thinking, on a regular basis. • Also, it helps in providing with the support to handle the various tasks, which are being performed at a regular pace, with full confidence which possess the inclusion of doing the shopping as well as making up of the planning regarding the personal budget. And, still even after having a perception as to how basic the type or the kind of the math pretends to be, mechanics which is being related to the maths stays to be a kind of the mystery to almost all the people around as generally all the students are being told to possess a full concentration or the focus just on the answers.
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But, instead according to some professor, having a proper or sound understanding, as well as the knowledge of the basic math, possess an inclusion of way more than just attaining with the proper solution or the correct one. As it has the inclusion of the following: •
Having a proper or the sound understanding of the nature or the behavior of the various numbers and also the concepts being present behind the various concepts being related to the maths and also the various numbers. • Paying a proper of the close attention as well as maintaining with the focus to the steps which are actually step by step at the back of the various difficult as well as the distinct calculations. • Possession of the thought regarding what is actually being solved or calculated and also the reason behind solving a particular issue or the question in an explicit or the specific way. Therefore, there pertains an approach which is actually rounded really well to the various basics being related to the mathematics, and also which is considered to be a sure short manner or the way to strengthen the existing knowledge or the information being possessed by a particular person. It also is necessary to attain various novel skills so that people would be able to properly deal with the maths as well as having a confidential as well as the deftly approach to attain a proper solution in maths only. Thus, it is all being made available or accessible to the various people out there within the course which is quite engaging and is being related to the various professors, also possessing the various books related to the art of possession of the mastering within the basic maths or the fundamental of the maths. Having a utilization of the similar skills which are being related to the teaching regarding the presentation or providing the various people with inspiration and also the experience which has been brought in the various books of the maths by the various professors. In such books, the various mysteries or the secrets which are behind the various topics being related to the maths are pertaining and which the people are required to possess the knowledge of.
1.1.1. Explore All the Essential Areas of Basic Math It is generally being designed, or structures for the various people who are actually the lifelong learners and the factor of age doesn’t matter as the people of all ages are being included within this particular category.
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Thus, the possession of the mastering the various zeros being related to the fundamentals of the maths which are being related to the topics which all the people are actually interested gaining the knowledge of also should know: •
•
•
•
Doing the process of adding, multiplying, and dividing as well as dividing the various numbers, be it the whole numbers, the integers, the fractional number or the various negative numbers. Doing the process of transforming or conversion amongst the various function which are the percentages, the fractions and also the decimals are included in this. Calculating or solving the various major problem which are being related to the real life which tends to possess the involvement of the proportions as well as the various ratios. Performing the various tasks or doing the work along with the exponents being related to the various whole numbers and also the square roots.
1.2. PROOF AND MATHEMATICAL ARGUMENT The process of writing up the mathematics explicitly as well taking up full protection as well as doing the work carefully has actually increasingly becoming quite essential as well as important as that particular person goes into the depths of doing the calculations or solving the problems. The various arguments which are being related to the recent or the advanced mathematics, might be possessing with the structure related to the logic, which are actually quite complicated, and also this particular structure pertaining has to be clearly understood by the people out there and most importantly to the one who is basically writing up the solutions being provided to the general public. But, being specific, the mathematics which is actually in the written form, should be at least more than the list of the various equations which has been provided, i.e., make the existing or the pertaining relation or the connection between them much more explicit or much more clearly understandable. The various words are being utilized very carefully which are actually quite useful in expressing up the pertaining relationship, or the connection is actually being considered as to be of utter importance or are quite essential.
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For instance: the process of writing up the equation “x ≥ 1, ≥ x” is considered to be quite ambiguous or chaotic, but for all the equation where x ≥ 1, and someone is possessing ≥ x, helps in making quite explicit regarding what all has been intended by this particular thing. Thus, various styles or way of writing up the mathematical terminologies will certainly vary insignificantly. Therefore, one must be able to develop or inhibit one’s own style or way of making apt utilization of the various words along with the various other symbols, so that they could be able make up the various arguments as understandable as possible.
1.2.1. Implication The various notions or the rules or the regulations regarding the various implications are considered to be fundamental in any of the argument which is related with the mathematics. Thus, in case if there persist two statements, be it A and B, then, the meaning of A implies B (in symbols A ⇒ B) provides with the meaning or the description regarding whenever the statement is being considered to be true, the statement B also should be true. For instance: x = 2 implies x2 = 4. The various implications regarding the several of ways or the manner of doing things are also being indicated, which is: A implies B, A ⇒ B, B is implied by A, If A then B, B if A.
Thus, it becomes quite critical to understand or realize that “If A then B” and “If B then A,” generally tend to be described or mean distinctively or are two distinct things. Similarly, “If x = 2, then x2 = 4” is true, but, “If x2 = 4 then x = 2” is false (x might be −2). However, at times these pertaining two statements which are A and B, are both being implied by one another, in the manner under which it might be considered that both the A as well as the B are being equal or equivalent (in symbols A ⇔ B). Thus, there pertains many of the ways of writing this particular thing, which generally possess the inclusion of:
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A is similar or equal to B, A ⇔ B, A implies and is implied by B, A if and only if B, A if B, A is, thus, considered to be the necessary and sufficient condition for B.
1.2.2. Proof The proof or the evidence is generally considered to be or being acknowledged as an argument, being quite careful and aware, which helps in incorporating or establishing the novel fact as well as the theorem which would be providing with the aid regarding the already provided various assumptions or the various hypothesis. Thus, much of the various types or the kinds of the proofs or the evidences, out of which some are being mentioned in here. Proof by deduction: A deductive proof consists of a sequence of statements or sentences each of which is deduced from previous ones or from hypotheses using standard mathematical properties. The final statement may be called a theorem. For example: Theorem 1.1: If x2 −3x + 1 < 0 then x > 0. Deductive proof: Having an assumption that x2 −3x + 1 < 0 (when the sign of the inequality has been changed), which provides with the implication regarding the inequality that 3x > 1, (since x 2 ≥ 0). Thus, it generally tends to follow that x > 13 (by having them divided, so, x > 0 (in accordance with the property of the order). Also, it has been noted that within this particular argument, each, and every step might be opted out from the one being present at previous stage with the help of the fact which is based on the standard mathematics. However, all of these steps are generally not reversible anyhow, which means one cannot reverse the order of the provided steps. Proof by contradiction: At times, it becomes quite simpler as well easy to possess an argument or the debate by the various contradictions, i.e., having an assumption regarding the conclusion or the outcome, which is actually being desired, tends to be false and thus, helps in deriving the various contradictions to the various facts which are actually being known to everyone. Theorem 1.2: If x2 −3x + 1 < 0 then x > 0. Proof by contradiction: An assumption has to be made that x2 −3x + 1 < 0 and also make a supposition that x ≤ 0.
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Then, x2 < 3x−1 ≤ 3×0 − 1 (rearranging and using x ≤ 0), so x2 < −1, which actually provides with the contradiction regarding the squaring up of the real number is generally considered to be as non negative (which includes 0 as well). We conclude that x > 0. Counterexamples: Thus, in order to present or show up with the statement which is actually false, it would be considered to be sufficient to provide with the single instance for which it is being applicable or does not get hold, are known as the counterexample. For example, the statement “x2 − 4x + 1 > 0 for all x > 0” is false. Thus, to possess a particular view or the note regarding the equation 22 − 4×2 + 1 = −3 ≤ 0, so that x should be equivalent to the numeric value 2, is thus, being the counterexample according to the statement which has been provided. Therefore, being quite specific, and also in order to demonstrate or determine the presence of the falsity, there persists no requirement or the need to sort or solve the pertaining inequality.
1.2.3. Mathematical Induction The method of the mathematical induction is generally being considered to be much more sophisticated which is actually related to the proof which can actually be deductive, and also which are being utilized in order to extract or derive out the various formulae as well as the facts, in accordance with the mathematics. Thus, induction is considered to be the method for the purpose of proving the various statements, which generally involves the various natural numbers which are 1, 2, 3 and so on. Thus, this particular idea is considered to be: •
A person would be able to provide with the proof regarding the statement when the value of n is equivalent to 1. • Also, in order to present or show that in case, the provided statement comes out to be true for some of the integers, then it is a mandate for it to be true for all the integers which are present immediately above it. Thus, with the help of this, a conclusion can be found out that the provided statement tends to be true for all the value n = 1, 2, 3, and so on. Therefore, formally, it is the principle of mathematical induction.
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Thus, let P(n) be a statement being dependent on the arbitrary positive integer n. Let us take a supposition that the following two steps can be undertaken: • •
The first one will be to verify that the value of P (1) is true; and In case of all the positive integers, which is n, it has to be shown that in case the value of the P (1) is true, then for P (n + 1) is a mandate to be true. Then, the statement of P (n) proves out to be true in case of all the positive integers n. And, this particular statement of P (n) is said to be the inductive hypothesis, wherein the first and the foremost step is known as the starting of the induction or the anchor. And also, the second step is said to be the inductive step. It has been noted that the principle of the induction is quite obvious intuitively, i.e., in case the value of the P (1) is true, and also the value of P(n) is equivalent to that of the value of P(n + 1), for all the values of n = 1, 2, 3 and so on, then, P (1) ⇒ P (2) ⇒ P (3) ⇒. ⇒ P(n) ⇒. By applying (2) with n = 1, 2, 3, and so on, in turn, so P(n) is true for all the values of n. Example: (Summation of series). For all the positive integer n
n(n + 1) 1+2+.+n= 2 Proof: Using the principle of induction. Let P (n) be the statement: 1+2+.+n= Then 1 = induction.
n(n + 1) 2 1(1 + 1) so, P(1) holds, which helps in starting up with the 2
Now an assumption has to be taken regarding the value of P(n) is true for some positive integer n. We relate the sum in P(n + 1) to that in P(n): 1 + 2 + . + n + (n + 1) = (1 + 2 + . + n) + (n + 1) =
n(n + 1) + (n + 1), (using P (n)) 2
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= n(n + 1) + 2(n + 1) 2
(n + 1) + (n + 2)
n + 1(n + 1) + 1
= = which is the statement of the value P(n + 1), 2 2 having completed with the step of inductive. Therefore, with the utilization of the Principle of Induction P(n), which is considered to be true for all positive integers n.
1.3. SETS, RELATIONS, AND FUNCTIONS In the daily life, various relations or the connection can be observed amongst the various objects. Thus, the context or the concept of this particular connection or the relation got developed or structured in the form of mathematics. This particular function of the word was first being introduced by the Leibnitz, in the year 1694. Function is considered to be the special kind or type of the relation. Thus, each and every function is generally is being acknowledged as the relation but not each and every relation is a function. Various functions, the basic definitions and also the various operations, which tends to possess an involvement of the sets, the Cartesian product of the two sets, the pertaining relation or the connection between the two sets.
1.3.1. Some Standard Notations In case of all the collections which are being written on the left hand side of the line which is vertical, the terms viz the tallness, the honesty, the intellectuality, are all not being clearly described or defined. Also, it is being considered to be the fact that all these particular notions or the rules or regulations generally tend to vary from person to person. Hence, all those collections which have been mentioned above, cannot be acknowledged as the sets. Thus, in case a collection is a set, then all the objects of that particular or specific collection is being represented as the element of this set. Generally, the set is being denoted by the capital letters and all the elements or the components, which are being inhibited, are all being denoted by the small letters.
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For instance, A = Toy elephant, packet of sweets, magazines. Following are some standard notations to represent sets: N: the set of natural numbers; W: the set of whole numbers; Z or I: the set of integers; Z + : the set of positive integers; Z–: the set of negative integers; Q: the set of rational numbers; R: the set of real numbers; C: the set of complex numbers. Other frequently used symbols are: ∈: ‘belongs to’;
∉: ‘does not belong to’; $: there exists;
$: there does not exist.
For instance, N is considered to be the set of the various natural numbers and also it is being quite known that 2 is a natural number but taking up the value of –2 would not be called as the natural number. Thus, this particular thing can be written in terms of symbolic form, i.e., 2 N ∈ and –2 N ∉.
1.3.2. Representation of a Set
Basically, there persist two methods, which help in providing with the representation of the particular set. •
Roster method (Tabular form): Within this particular method of the set which is being represented by the process of listing of all the various pertaining elements or the components, which provides with the aid in segregating them using the commas and also by enclosing them into the curly brackets. Thus, in case, V is considered to be the set of vowels of the alphabet in English, it might be quite useful in writing these in the Roster format as: V = {a, e, i, o, u}
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In case, the value of A is considered to be the set of natural numbers which is less than 7. Then A = {1, 2, 3, 4, 5, 6}, is said to be in the Roster format. •
Setbuilder form: Being in this format, the various elements or the components of this particular set, are not being listed but instead, they are being presented by the property which is quite common to all. Let the supposition by taking up the value of V, which is considered to be the vowel of the alphabets in English, then the value of V can be presented in the set builder format as: (i) (ii)
V = {x: x is a vowel of English alphabet} Let us consider A to be the set of natural numbers, which are less than 7. Then A = {x: x ∈ N and 1 x £ , which follow according to the usual conventions. For example: a < b means a ≤ b and = b
1.4.2. Integers Someone has to be begun with somewhere, and within this particular course, one has to start up from the various natural numbers and thus, helps in creating or making up of the various other systems of the number from that particular point. Thus, it might be possible to take up more of the basis of the starting point, which is being provided with the name of Peano Axioms, and helps in developing the various natural numbers and the various integers from that particular point. Therefore, an assumption will be taken into consideration that various integers Z = {..., −3, −2, −1, 0, 1, 2, 3.} are being ‘familiar’ and also ‘wellunderstood.’ Also, being specific, the assumption has to be taken into acknowledgment regarding the various properties of the addition, multiplication as well as the order. Thus: Proposition: The integers Z form a particular ring being present under the usual process of addition as well as the multiplication, and this particular ring is being ordered under ≤. The integers are being adequate for many of the provided things, in specific, for the various purposes of the process of the induction of mathematics. However, these are being present in some of the provided ways which are being limited, and notably that integers do not possess multiplicative inverses, or in the other words, that they do not create or form up a particular field.
1.4.3. Rational Numbers The question arises regarding the requirement of the various rational numbers and also the reason regarding the non satisfaction of the people with their particular intuition regarding them, as it was being pretended to be connected or being related with Z.
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Thus, it is just about the possibility or could. And, this particular difficulty or the complexity being aroused relating to the various rational numbers, are connected with the various distinct or the different formats, viz: say, 3/5 and 6/10, and in real these are generally being presented or represented by the like or the same rational number. Thus, it can be witnessed or quite visible regarding how the language of equivalence relations can be properly utilized so as to overcome the particular complexity or the difficulty. But, also this will be a kind of the form of warmup exercise for the process of making up or the construction of real numbers from the various rational numbers – where the intuition of a particular person is being considered much flakier. A rational number, say 3/5, is only or just a fancy notation for a pair of the ordered number of the pertaining integers (3, 5), and there pertains the requirement or the need so as to identify or recognize this with the provided pairs (6, 10), (9, 15), etc., which are being the distinct or the different presentation or the representation of the like or the same fraction. Lemma R is an equivalence relation. Proof: Utilization of the various properties of the process of multiplication of the provided integers: (R) a*b = b*a ⇒ (a, b) R (a, b).
(S) (a, b) R (c, d) ⇒ a*d = b*c ⇒ c*b = d*a ⇒ (c, d) R (a, b). (T) (a, b) R (c, d) & (c, d) R (e, f) ⇒ a*d = b*c & c*f = d*e ⇒ a*d*f = b*c*f = b*d*e ⇒ a*f = b*e (as = 0) ⇒ (a, b) R (e, f)
Definition: The equivalence classes of R are generally being called the rational numbers: Q = {[(x, y)]: (x, y) ∈ X}.
The equivalence class [(a, b)] is being denoted by or a/b.
1.4.4. Real Numbers The various real numbers are being thought or considered as the process of filling up the various holes that has already been noted or identified in Q. Thus, a particular clue has been provided as to how to provide with the definition a particular real number: one has to recognize or identify it
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particularly along with the set of all of the provided rational numbers which are being provided in the following and for which it will be acting as the least upper bound). Definition: A Dedekind cut or cut is being a nonempty subset A ⊆ Q along with the following properties: (C1) A is bounded above, i.e., it possesses an upper bound; (C2) A has no maximum; (C3) A is being closed downwards, that is in case, x ∈ A and y ≤ x, then y ∈ A.
The set of all Dedekind cuts is being provided as or being denoted by R and termed the various real numbers. Example: {x ∈ Q: x < 1} is a Dedekind cut, {x ∈ Q: x ≤ 1}, {x ∈ Z: x < 1} and Q are not.
Thus, in accordance with the building or creating up of the theory of real numbers, there pertains the requirement or the need of the following steps: (1. (2. (3. (4.
Define the basic operations + and · on R. Define ≤ on R. Identifying or recognizing the particular copy of Q in R. Establishment of the various basic properties of the operations and ≤, that is, it has to be shown that R is an ordered field. (5. Prove that R is complete. Implementation or the execution of the whole of the program in proper details is instead quite long as well as tedious, and it happens generally due to the technical complexity or the difficulty of performing the process of the multiplication of the various numbers which are actually negative. Thus, this particular process has been sketched paying a proper attention to the points which are quite essential or are of utter importance.
1.4.5. Complex Number Taken from the description provided by the product of the various cuts, it is quite evident that ≥ 0, for all x ∈ R, so it generally tends to follow that all the numbers which are negative, do not possess the square roots or even in case of any of the roots which are even. Thus, specifically, there pertains no number x ∈ R, which will help in satisfying x^2 = −1. Therefore, a number can be invented similar to this, and it could be adjoined with the R.
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This basically means that creation or the making up of the various complex digits. But, the something which might of utter importance would be happening as it would have turned or transformed into all of the real numbers which all generally possess the roots. It is quite better that n z generally tend to pertain even in the case of the various numbers which are complex in nature which is n, which means that all of the numbers which are complex generally tend to possess the roots. Even better than that, all of the existing polynomial equation P (z) = 0, where P is considered to be a polynomial with the various complex coefficients, which all provide with a particular solution in the format of the complex numbers. And this particularly last result is being described as the Fundamental Theorem of Algebra. In comparison to the construction of all the real numbers from that of the rational, or even the construction or the making up of the various rational numbers from integers, and also the construction of the various complex numbers from real is considered to be quite straightforward. Definition C = R×R = {(x, y): x, y ∈ R}.
In other words, the various complex numbers are being considered to be the ordered pairs of real. Addition and multiplication are being defined as follows:
(a, b) + (c, d) =(a + c, b + d), (a, b) ⋅ (c, d) =(ac− bd, ad + bc) Proposition: For all z, w ∈ C: • •
z*u = zw; z + w ≤ z + w. (The triangle inequality);
Pr oof . (i) Let z = a + bi, w = c+ di.We have  zu 2 = (ac− bd) + (ad + bc) i 2 = (ac− bd) 2 + (ad + bc) 2 (ii) We have  z + w 2 = (z + w)( z + w) =+ (z w)(z + w) =zz + (zw + wz) + ww =  z 2 +2 Rw(zw)+  w 2 (where Re(z) = a if z = a + bi) ≤ z 2 +2  zw  +  w 2 ≤ z 2 +2  z  w  +  w 2 = ( z  +  w ) 2 .
(using (i))
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1.5. SOME NUMBER THEORY 1.5.1. Prime Numbers • Prime Factorization: The set of natural numbers is: N = {1, 2, 3, 4}, And the set of integers is: Z = {. −2, −1, 0, 1, 2.}. Definition (Divides): In case the values of a, b ∈ Z we say that a divides b, written a  b, if a*c = b for some value of c ∈ Z. Then, in that case, it is being said that a is a divisor of b. We say that a does not divide b, provided that a – b, if there is no c ∈ Z such that a*c = b.
For example, someone possesses 2  6 and −3  15. Also, all of the integers tend to divide 0, and 0 divides only 0. However, 3 does not divide 7 in Z. The number 1 is neither said to be the prime nor the composite. The first few primes of N are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, and the first few composites are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34. •
The Greatest Common Divisor: Generally, the various pertaining notions are being utilized as one of the greatest common divisors of the two pertaining integers to prove that in case p is a prime number and p  a*b, then p  a or p  b. Proving this is the key step in our proof of Theorem 1.1.6. Definition (Greatest Common Divisor): Let gcd (a, b) = max {d ∈ Z: d  a and d  b}, unless both of the values of a and b are 0, in which case gcd (0, 0) = 0. For example, gcd (1, 2) = 1, gcd (6, 27) = 3, and for any of the gcd (0, a) = gcd (a, 0) = a If a ≠ 0, the greatest common divisor pertains because in case the value of d  a then d ≤ a, and also, there are only a positive integers ≤ a. Similarly, or likely, the gcd exists when b ≠ 0.
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1.5.2. Continued Fractions The golden ratio (1 + √ 5)/2 is equivalent to that of the infinite fraction
and the fraction 103993 = 3.14159265301190260407... 33102 is considered to be an excellent approximation to π. Thus, both of them are being considered to be the observations are being explained by the various continued fractions. Continued fractions are generally being theoretically beautiful and provide with the various tools or the equipment that attains or yield the various powerful algorithms for the purpose of solving the pertaining problems in case of the number theory. For example, in case of the continued fractions which tends to provide with a fast way to write a prime—even a hundred digit prime—as a particular sum of two squares, when it is being possible. Continued fractions are thus, being considered to be the beautiful algorithmic and conceptual tool in case of the number theory that might possess many of the applications. For example, they might be providing with a surprisingly efficient manner to recognize or identify a rational number which is provided to just the first few digits of its decimal expansion, and they provide with the sense in which e is being considered to be “less complicated” as compared to π (Figure 1.1).
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Figure 1.1: Graph of a continued fraction. Source: https://wstein.org/ent/ent.pdf
•
Finite Continued Fractions: This section is all about the various continued fractions being of the form [a0, a1., am] for some of the values of m ≥ 0. An inductive definition of numbers pn and qn is being provided or given such that for all the values of n ≤ m [a0, a1., an] =
pn . qn
Then the various results are being provided which are being related to the
pn qn
pertaining formulas for the determinants of the 2 × 2 matrices pn and qn
pn − 1 qn − 1
pn − 2 qn − 2 which we will be repeatedly utilized in order to deduce the
various properties of the sequence of the partial convergent such that [a0, ak]. •
Infinite Continued Fractions: Thus, this particular section generally tends to begin along with the procedure of the continued fraction, which is being associated with a sequence a0, a1 including of all the integers to a particular real number x. After providing with the various examples, it can be proven that x = limn→∞ [a0, a1, an] by proving that all the odd and also the even partial convergent tend to become arbitrarily close to each other.
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It can also be shown that in the case of a0, a1 is an infinite sequence of positive integers, then the sequence of cn = [a0, a1, an] tends to converge.
More specifically, in case, an is an arbitrary sequence of positive reals such that an diverges then (cn) converges.
1.6. CASE STUDY: ANALYSIS ON MATHEMATICS FUNDAMENTAL KNOWLEDGE FOR MATHEMATICS ENGINEERING COURSES BASED ON A COMPARATIVE STUDY OF STUDENTS’ ENTRY PERFORMANCE 1.6.1. Introduction Mathematics is a significant topic supporting a large number of engineering courses, and consequently, it is important for engineering students to hold a strong mathematics fundamental knowledge that can keep their motivation for equitable progress of their engineering programmers. Pyle (2001) stated that engineering as a profession requires a clear understanding of mathematics, sciences, and technology and Sazhin (1998) mentioned that engineering graduate acquires not only a practical but also an abstract understanding of mathematics. Therefore, it is crucial that in university level, most of the programs of study require mathematics, at which the ability to master mathematical skills are important indicator of potential for students’ in all levels of academics endeavors (Tang et al., 2009). Lawson (2003) found that changes in basic mathematical knowledge have direct effect to many mathematical skills that are essential for those undergraduate degree courses with a significant mathematical content. Meanwhile, a study done by Lawson (1997) and Stephen et al. (2008) revealed that there has been declining in the ability of students in certain fundamental mathematical topics recently. This is the consequence of students’ prior experiences and knowledge they earned from the preuniversity learning process.
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This study is a part of action research project to investigate the students’ performance in mathematics engineering courses in FKAB, UKM, and extension of the papers done by Zainuri et al. (2009) and Othman et al. (2009), which are presented during the Congress of Kongres Pengajaran dan Pembelajaran UKM 2009 at Langkawi, Kedah. Thus, this study focuses on the students’ entry performance from 2008 to 2010 (and onward) based on mathematics pretest. The objective of this study is to identify the mathematical topics which are considered difficult and challenging by the firstyear engineering students. This paper also discusses the students’ performance based on genders, specifically in which mathematics topics, male or female students performed better than others.
1.6.2. Methodology A set of mathematics pretest were given to the firstyear students from academic’s session 2008/2009 to academic’s session 2009/2010 at Faculty of Engineering and Built Environment (FKAB), UKM during the second week of each semester. Questions on the PreTest covered on elementary mathematical concepts such as functions and graphs, differentiation, integration, vectors, and etc. A total number of 327, 296 and 232 students from academic’s session 2006/2007, 2008/2009 and 2009/2010, respectively, are involved in this study. Results were analyzed on the performance based on genders and topics.
1.6.3. Results and Analysis The student from session 2008/2009 obtains a better result with an average of 74.44% compared to session 2009/2010 with 67.90%. In the range of 80–100, students from session 2008/2009 performed better than students from session 2009/2010 with 48.3% and 33.2%, respectively (Figure 1.2). For the lowest range, that is 20–40, the student from session 2008/2009 once again shows a better result where only 3% in this range as compared to session 2009/2010.
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Figure 1.2: Pretest result based on students’ entry performance from academic’s session 2008/2009 to 2009/2010. Source: https://www.sciencedirect.com/science/article/pii/ S1877042812038475
The pretest result is also compared across gender as shown in Figures 1.2 and 1.3. Overall, male students performed better compared to female students from academic’s session 2008/2009 to 2009/2010. It is found that both male and female students from session 2008/2009 get the highest average result, which is 77.44 and 69.97, respectively.
Figure 1.3: Pretest result for male students from academic’s session 2008/2009 to 2009/2010. Source: https://www.sciencedirect.com/science/article/pii/ S1877042812038475
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1.6.4. Discussion The intake from academic session 2009/2010. Even, the intake from academic’s session 2008/2009 did well compared to 2009/2010 for both male and female students for the pretest performance. Comparison between genders for academic’s sessions 2008/2009 and 2009/2010, obviously shows that male students’ performance better in pretest compared to female students. Based on the results in this study we can agree with the findings found by Lawson (2003), that there has been a significant decline in many mathematical skills that are regarded by higher education as essential for those undertaking degree courses with significant mathematical content such as engineering programmed. This scenario can be observed in the comparison between intake from academic’s session 2008/2009 and 2009/2010 at FKAB, UKM based on mathematics topics, which obviously shown that there were a decrease in the percentage of answering pretest questions correctly. The factors of the declining in the mathematics performance of newly intake students for academic’s session 2009/2010 might be caused by various reasons. Holes et al. (2001) mentioned that the role of higher education in the UK, particularly has changed significantly. At the same time, the numbers of students accepted to university increased substantially due to the change in education policy of the government and also the aims and objectives of a university engineering degree programmed have undergone a dramatic change in order to fulfill employers’ expectation and demands. Meanwhile, Hill et al. (2005) study revealed that teachers’ mathematical knowledge was significantly influenced students’ performance. Thus, this finding provides support for the government to design policy in order to improve students’ mathematics performance by improving teachers’ mathematics knowledge. Consequently, lack of academic skills, such as essay writing and note taking, lack of selfmanagement skills, lack of confidence in generic skills and poor time management (Goldfinch and Hughes, 2007) are among the factor contribute to the declining performance of year one students.
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1.6.5. Conclusion This paper revealed students from academic session 2008/2009, for male and female, achieved better performance in the PreTest than students from academic session 2009/2010. However, the students from both academic’s sessions and genders performed poorly in the topics of Vector and followed by Limit and Continuity. These findings are absolutely alarming for lecturers in FKAB, UKM since these topics are the main mathematical contents required in engineering courses. Therefore, with these findings and understanding, lecturers need to take proactive action in order to understand the problems faced by students in an effort to improve student performance by creating an appropriate curriculum for firstyear students and determine whether students would benefit from additional support. On the other hands, since these students are the end product of Ministry of Education (MOE), thus MOE should review the teaching and learning method and syllabus for the mathematics courses in Matriculation, STPM, Assai Sains, Fundamental Study, and Diploma. This is an effort to enhance students’ mathematics fundamentals knowledge which is required in pursuing undergraduate engineering study.
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REFERENCES 1.
Apexlearning.com (2018). Fundamental Math: Apex Learning. [online] Available at: https://www.apexlearning.com/course/736 [Accessed 20 August 2018]. 2. Burzynski, D., & Ellis, W., (2018). Fundamentals of Mathematics, p. 724. 3. English, (2018). Mastering the Fundamentals of Mathematics. [online] Available at: https://www.thegreatcourses.com/courses/masteringthefundamentalsofmathematics.html [Accessed 20 August 2018]. 4. Falconer, K., (2010). Fundamentals of Pure Mathematics (p. 49). [eBook]. Available at: http://wwwmaths.mcs.standrews. ac.uk/~kenneth/FundPureNotes.pdf [Accessed 20 August 2018]. 5. Gies, T., (2018). The science direct accessibility journey: A case study. Learned Publishing, 31(1), pp. 69–76. 6. Glaister, P., & Glaister, M. E., (2018). Mathematical Argument, Language and Proof (p. 4). [eBook]. Available at: https://www.ma. org.uk/resources/2017SepProofGlaistersMiS.pdf [Accessed 20 August 2018]. 7. Kumar, N., (n.d.). Construction of Number Systems (p. 31). [eBook]. Available at: https://www.math.wustl.edu/~kumar/courses/310–2011/ Peano.pdf [Accessed 20 August 2018]. 8. Schroder, B., (2018). Fundamentals of Mathematics (p. 49). [eBook]. Available at: http://www.math.usm.edu/schroeder/slides/fund_slides/ fund_intro.pdf [Accessed 20 August 2018]. 9. Sets, Relations, and Functions. (n.d.). [eBook] p. 44. Available at: http://download.nos.org/srsec311new/L.No. 15A.pdf [Accessed 20 August 2018]. 10. Stein, W., (2017). Elementary Number Theory: Primes, Congruence, and Secrets (p. 172). [eBook]. Available at: https://wstein.org/ent/ent. pdf [Accessed 20 August 2018].
2 Algebra and Basic Math
CONTENTS 2.1. Fundamentals Of Algebra ................................................................. 38 2.2. Operations On Monomials and Polynomials..................................... 40 2.3. Linear Equations In One Variable ...................................................... 46 2.4. Problems To Solve............................................................................. 58 References ............................................................................................... 64
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Mathematics is a vast subject, which includes arithmetic, calculations, statistics, and various other branches. One of the significant fields that make up the great subject of mathematics is ‘Algebra.’ Algebra is the basis for all the topics that come up in the advanced form of mathematics like calculus, relations, and functions and other important topics. Hence it becomes important for the readers and learners to revise and view the basic concepts of algebra before moving forward.
2.1. FUNDAMENTALS OF ALGEBRA There is a very significant part of mathematics which is made up by numbers and some general rules of arithmetic and that part is commonly known as ‘algebra.’ Algebra takes the methods and learnings of arithmetic in such a way that it becomes easy to imply the rules related with the calculation of numbers and make use of these rules to work with some symbols too other than the numbers. The adoption of algebra provides easy access to several other branches of mathematics, rather than an abrupt makeover into new fields, with the use of previously attained knowledge of the use of basic arithmetical operations. The way of writing quantities in some common ways instead of a particular set of arithmetic terms is widely known. A very common example of such representation is the formula of the perimeter of a rectangle, which may be written as: P = 2L + 2W Where the letter P represents the perimeter of the rectangle, the letter L represents the length and the letter W represents width. It should be noted that 2L = 2 (L) and 2W = 2 (W). Had the letters L and W been numerical values, it would have been important to use the brackets or a multiplication sign, but in this case, the meaning of the term like 2L is distinct, without any extra use of symbols or signs. Every formula is an expression in algebra, though they may not be seen in that manner. The letters that are made use of in these expressions of algebra are known as ‘literal numbers.’ Another place where literal numbers may be used is the expression of the mathematical rules of action. These laws or rules, as may be said, are discussed hereafter.
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2.1.1. Commutative Laws Commutative laws are the ones that hold true even after any rearrangement is done on either side. They may be illustrated as shown below. Addition: The expression for addition in algebra in which commutative law applies can be written as: a+b=b+a •
This law may also be applied for more terms as in: a + (b + c) = a + (c + b) = (c + b) + a Thus, this law establishes that “the sum of two or more addends is the same regardless of the order in which the addends are arranged.” •
Multiplication: The way in which the commutative law for multiplication may be written in the algebra is:
ab = ba Thus, this law may states that “the product of two or more factors is the same regardless of the order in which the factors are arranged.”
2.1.2. Associative Law The associate law may be applied to the operation of addition and multiplication where the terms and factors may be grouped together in arithmetic or algebraic expression. •
Addition: The form in which the associative form for addition may be illustrated in algebra is: a + b + c = (a + b) + c = a + (b + c) Thus, this law states that the sum of three or more addends may remain the same irrespective of the manner in which they are grouped. •
Multiplication: The form in which the associative method for multiplication may be illustrated in algebra is: a·b·c = (a·b)·c = a·(b·c) Thus, this associative law for multiplication states that the product of three or more factors may remain the same irrespective of the way in which the factors may be grouped.
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2.1.3. Distributive Law The distributive law indicates to the arrangement of factors in a distributive way among the terms of an expression meant for addition. The form in which this law may be written as, in the algebra, is: a (b + c) = ab + bc Thus, this law of distribution states that “if the sum of two or more quantities is multiplied by a third quantity the product is found by applying the multiplier to each of the original quantities separately and summing the resulting expressions.
2.1.4. The Algebraic Expression An algebraic expression is composed of various signs and symbols that are used in algebra. These consist of Arabic numerals, literal numbers, the signs of operation, and such more things. Such an expression eventually denotes just one number or one value. Hence, just like the addition of 5 and 8 gives one value as 13, the expression c + d will also yield one final value. There may be, sometimes, a formation of longer expressions by using various signs and symbols, but eventually, they will result in only one number, no matter how long the expression may seem.
2.2. OPERATIONS ON MONOMIALS AND POLYNOMIALS 2.2.1. Multiplication of Monomials If a monomial such as 3abc is to be multiplied by a numerical multiplier, for example, 5, the coefficient alone is multiplied, as in the following example: 5 x 3abc = l5abc. When the numerical factor is not the Initial factor of the expression, as in x(2a), the result of the multiplication is not written as x2a. Instead, the numerical factor is interchanged with literal factors by use of the commutative law of multiplication. The literal factors are usually interchanged to place them. In alphabetical order, and the final result is as follows: x(2a) = 2ax. The rule for multiplication of monomials may be stated as: •
Multiply the numerical coefficients to form the coefficient of the product.
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•
Multiply the literal factors, combining exponents in like factors, to form the literal part of the product. The complete process is illustrated in the following example: (2ab) (3a2) (2b3) = 12a1 + 2 b1 + 3 = 12a3b4.
2.2.2. Division of Monomials As may be expected, the process of dividing is the inverse of multiplying. Because 3 x 2a = 6a, 6a/3 = 2a, or 6a/2 = 3a. Thus, when the divisor is numerical, divide the coefficient of the dividend by the divisor. When the divisor contains literal parts that are also in the dividend, cancellation may be performed as in arithmetic. For example, 6ab + 3a may be written as follows: (2) (3a) (b) / 3a Cancellation of the common literal factor, 3a, from the numerator and denominator leaves 2b as the answer for this division problem. When the same literal factors appear in both the divisor and the dividend, but with different exponents, cancellation may still be used as follows:
14a 3b3 x (7)(2) a 2 ab3 x = −21a 2b5 x (7)(−3) a 2 b3b 2 x 2a 2a = = − 2 2 −3b 3b This same problem may be solved without thinking in terms of cancellation, by rewriting with negative exponents as follows:
14a 3b3 x 2a 3− 2b3−5 x1−1 = −21a 2b5 x −3 −2 2ab 2a = = −3 −3b 2 2a = − 2 3b
2.2.3. Multiplication of Two or More Polynomials As with the monomial multiplier, we explain the multiplication of a polynomial by a polynomial by use of an arithmetic example. To multiply (3
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+ 2) (6–4), we could do the operation within the parentheses first and then multiply, as follows: (3 + 2) (6–4) = (5) (2) = 10 However, thinking of the quantity (3 + 2) as one term, we can use the method described for a monomial multiplier. That is, we can multiply each term of the multiplicand by the multiplier, (3 + 2), with the following result: (3 + 2)(6 – 4) = [(3 + 2) x 6 – (3 + 2) x 4] Now considering each of the two resulting products separately, we note that each is a binomial multiplied by a monomial. The first is (3 + 2) 6 = (3 x 6) + (2 x 6) and the second is –(3 + 2) 4 = –[(3 x 4) + (2 x 4)] = –(3 x 4) – (2 x 4) Thus, we have the following result: (3 + 2) (6 – 4) = (3 x 6) (2 x 6) – (3 x 4) – (2 x 4) = 18 + 12 – 12 – 8 = 10 The complete product is formed by multiplying each term of the multiplicand separately by each term of the multiplier and combining the results with due regard to signs. Now let us apply this method in two examples involving literal numbers. • (a + b) (m + n) = am + an + bm + bn • (2b + c) (r + s + 3t – u) = 2br + 2bs + 6bt – 2bu + cr + cs + 3ct – cu The rule governing these examples is stated as follows: The product of any two polynomials is found by multiplying each term of one by each term of the other and adding the results algebraically. It is often convenient, especially when either of the expressions contains more than two terms, to place the polynomial with the fewer terms beneath the other polynomial and multiply termbyterm beginning at the left. Like terms of the partial products are placed one beneath the other to facilitate addition. Suppose it is asked to find the product of 3x2–7x – 9 and 2x – 3.
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The procedure is,
3x 2 − 7 x − 9 2x − 3 6 x 3 − 14 x 2 − 18 − 9 x 2 + 21x + 27 6 x 3 − 23 x 2 + 3 x + 27
2.2.4. Division of Polynomial by Polynomial Division of one polynomial by another proceeds as follows: •
Arrange both the dividend and the divisor in either descending or ascending powers of the same letter. • Divide the first term of the dividend by the first term of the divisor and write the result as the first term of the quotient. • Multiply the complete divisor by the quotient just obtained, write the terms of the product under the like terms of the dividend, and subtract this expression from the dividend. • Consider the remainder as a new dividend and repeat steps 1, 2, and 3. Example: (10x3–7x2y – l6xy2 + 12y3)/(5x – 6y) Solution: 3
2 x 2 + xy − 2 y 2 2
10 x − 7 x y − 16 xy 2 + 12 y 3 5x − 6 y 10 x3 − 12 x 2 y 5 x 2 y − 16 xy 2 5 x 2 y − 6 xy 2 −10 xy 2 + 12 y 3 −10 xy 2 + 12 y 3 In the example just shown, it is begun by dividing the first term, 10x3, of the dividend by the first term, 5x, of the divisor. The result is 2x2. This is the first term of the quotient.
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Next, the divisor is multiplied by 2x2 and this product is subtracted from the dividend. Use the remainder as a new dividend. Get the second term, xy, in the quotient by dividing the first term, 5x2y, of the new dividend by the first term, 5x, of the divisor. Multiply the divisor by xy and again subtract from the dividend. Continue the process until the remainder is zero or is of a degree lower than the divisor. In the example being considered, the remainder is zero (indicated by the double line at the bottom). The quotient is 2x2 + xy – 2y2. The following long division problem is an example in which a remainder is produced:
x+3
x2 − x +3 2
x3 + 2 x 3
x + 3x
+5
2
− x2 − x 2 − 3x 3x + 5 3x + 9 −4 The remainder is –4. Notice that the term –3x in the second step of this problem is subtracted from zero, since there is no term containing x in the dividend. When writing down a dividend for long division, leave spaces for missing terms which may enter during the long division process. In arithmetic, division problems are often arranged as follows, in order to emphasize the relationship between the remainder and the divisor:
5 1 = 2+ 2 2 This same type of arrangement is used in algebra. For example, in the problem just shown, the results could be written as follows:
x3 + 2 x 2 + 5 4 = x2 − x + 3 − x+3 x+3 Remember, before dividing polynomials arrange the terms in the dividend and divisor according to either descending or ascending powers of one of the literal numbers. When only one literal number occurs, the terms
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are usually arranged in order of descending powers. For example, in the polynomial 2x2 + 4x + 5–7x the highest power among the literal terms is x3. If the terms are arranged according to descending powers of x, the term in x3 should appear first. The x3 term should be followed by the x2 term, the x term, and finally the constant term. The polynomial arranged according to descending powers of x is 4x3 + 2x2–7x + 5. Suppose that 4ab + b2 + 15a2 is to be divided by 3a + 2b. Since 3a can be divided evenly into 15a2, arrange the terms according to descending powers of a. The dividend takes the form 15a2 + 4ab + b2.
2.2.5. Factorizing the Algebraic Expression A factor of a quantity N is an expression which can be divided Into N without producing a remainder. Thus 2 and 3 are factors of 6, and the factors of 5x are 5 and x. Conversely, when all of the factors of N are multiplied together, the product is N. This definition is extended to include polynomials. The factors of a polynomial are two or more expressions which, when multiplied together, give the polynomial as a product. For example, 3, x, and x2–4 are factors of 3x3–12x, as the following equation shows: (3) (x) (x2–4) = 3x3–12x The factors 3 and x, which are common to both terms of the polynomial 3x –12x, are called common factors. The distributive principle is an important part of the concept of factoring. It may be stated as follows: 3
If the sum of two or more quantities is multiplied by a third quantity, the product is found by applying the multiplier to each of the original quantities separately and summing the resulting expressions. It is this principle which allows us to separate common factors from the terms of a polynomial. Just as with numbers, an algebraic expression is a prime factor if it has no other factors except itself and 1. The factor x2–4 is not prime, since it can be separated into x – 2 and x2. The factors x – 2 and x + 2 are both prime factors, since they cannot be separated into other factors. The process of finding the factors of a polynomial is called factoring. An expression is said to be factored completely when it has been separated into its prime factors. The polynomial 3x3–12x is factored completely as follows: 3x3–12x = 3x (x – 2) (x + 2).
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2.3. LINEAR EQUATIONS IN ONE VARIABLE One main purpose, out of the many, for a concentrated learning of polynomials, factoring, fractions, and grouping symbols is to get equipped with the skills to solve an equation. The equation may be considered as the most significant aspect of the algebra and the better knowledge a student possesses of the methods of dealing with these equations, and the more will be their chances of solving them. It is important to know about the words that may be utilized in the conversation about the equations, prior to getting to know how to solve them. An Equation may be referred to as a statement in which two expressions may be expressed as equal in value. Thus, the following are known as equations: 3 + 7 = 10 and V = L × B × H (Volume of a rectangle = Length × Breadth × Height) The portion to the left side of the sign of equality is known as ‘left member’ or ‘first member’ of the equation. The one to the right side of the sign is known as ‘right member’ or ‘second member’ of the equation. The two members of the equation may be equated with the two weights that may be used on a balance scale. This analogy may be used to teach students who are beginning to learn about the equations. In the analogy of the scale, it may be understood that any kind of addition or subtraction on either side may bring about an equal alteration on the other side. If this does not happen, the scale may not be balanced. This same logic governs the solving of equations, in which to maintain the equality, the changes must be made accordingly.
2.3.1. Constants and Variables The statements in algebra comprise of two elements: • constant; and • variables. A constant is an element that maintains its value as same during a certain problem. A variable is an element that may have its value varying or changing, during the problem. The constants may be of two kinds: • •
fixed; and arbitrary.
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The fixed constants may be any number such as 4, 8. 13, 3/5, or 3.8. The value of fixed constants does not vary ever. For example, in the equation, 8x + 9 = 2, where the numbers 8, 9 and 2 are fixed constants. Arbitrary constants are those that vary with respect to the problems and may be given values that may not be alike in all the problems. Arbitrary constants may be denoted by letters that may generally be picked from the starting of the alphabet like a, b, c or d. For example, in the following equation ax + b = c, where the letters a, b, and c are the arbitrary constants. The expression ax + b = c may be indicating to a lot of linear equations. These equations may turn into fixed constant type, if a, b, and c are assigned some specific values like, for example, a = 8, b = 9 and c = 23, in which case the equation takes the form: 8x + 9 = 23 A variable, on the other hand, may possess one or more than one value in a conversation. The variables are generally denoted by the letters that come at the end of the alphabet like x, y, z or w, as may be the requirement of the expression. For example, in the expression 8x + 2, the denomination ‘x’ is the variable. The value of the expression changes with a change in the value of x. Say, if x = 3, then 8x + 9 = 8(3) + 9 = 24 + 9 = 33 And if x = 4, then 8x + 9 = 8(4) + 9 = 32 + 9 = 41 This may carry on like this for all the values of x as and when required and selected. If the statement 8x + 9 is equated with a specific value, for instance 25, then the equation that is formed 8x + 9 = 25 may hold true for only one value of x. In this equation, the value of that variable is 2 as, 8(2) + 9 = 25 In a statement in algebra, the elements that comprise of variables are known as ‘variable terms.’ And the elements that do not consist of any
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variable are known as ‘constant terms.’ The statement 8x + 9 comprises of one variable term and one constant term. The term that is variable is 8x, and the one that is constant is 9. In another expression bx + c, bx is a term that is variable, and c is the one that is constant. A variable term is generally indicated by naming the variable it comprises of. For example, in 8x + 9, 8x is the xterm and in an expression like cx + dy, cx is the xterm whereas dy is the yterm.
2.3.2. The Degree of An Equation The degree of an equation may be referred to as the exponent of the highest power to which the variable of the equation, that contains a maximum of one variable with every term, is raised in the expression. For example, in the equation, 2x + 6 = 12 The highest exponent of the highest power of x is 1 and hence the equation is a ‘firstdegree equation.’ In another example of an equation, x2 + 6x = 50 the exponent of the highest power is 2 and hence the equation is a ‘seconddegree equation.’ According to one more equation, x5 + 7x + 3x2 = 83 the exponent of the highest power is 5 and therefore the equation is a ‘fifthdegree equation.’ The equation of the form, 9x + 8y = 56 is an equation of first degree in two variables. In an equation where more than one variable come together in a term in the expression, like in x2y5 = 72, the degree of the equation may be found by adding up the exponents of the powers and the highest among them is thereafter, the degree of the equation. For example, in the given equation, the exponents 2 and 5 are added to give the result of 7 and the equation may be termed as one of seventh degree.
2.3.3. Linear Equation To provide the learners with a visualization of some specifically related facts, the graphs may be used in several different ways. For instance, they may be used to present various trends in the businesses, the output from the production of some goods, constantly distinct accomplishments and many other such processes. Many kinds of graphs such as line graph, bar
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graph, circle graph, and other such kinds may be utilized according to the specific need of the process. The graphs in algebra too may be used for the purpose of visualization in the form of pictures to portray a huge amount of information about the equations. More often than not, the numbers may satisfy the layout of the equation, when they are substituted with the variables in the expression. On a specific kind of graph, many of these numerical values are plotted, and after enough numbers have been plotted, a line is made that goes through these points that may result in different types of curves for different equations. The shape of the equations, for one or two variables in the first degree, is a straight line. Hence, the name of this kind of equations is ‘linear equations.’ The shapes differ with the change in degree resulting in other kinds of shapes. The term ‘linear equations’ is now used to denote the equations of the first degree, irrespective of the number of variables that equation may comprise of.
2.3.4. Identities Any expression of equality that may be in one or more than one variable may either be an ‘identity’ or a ‘conditional equation.’ An identity can be referred to as an equality that establishes a fact, like in the following expressions: • 5 + 2 = 7; • 6m + 3m = 9m; • 5 (y + 2) = 5y + 10. It may be noted that the third equation simply portrays the factored form of 5y + 10 and gives true outcomes when any value of y is substituted. For instance, if y = 3, it turns into, 5 (3 + 2) = 5 (3) + 10 5 (5) = 15 + 10 25 = 25 In the case where y takes the value as a negative, such as –3, this identity turns into, 5 (–3 + 2) = 5 (–3) + 10 5 (–1) = –15 + 10 –5 = –5
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An identity is stated when the two sides of the equation are solved to come to the same expression or the same numerical value. In the case where y is substituted with 3, both the corresponding sides in the equation 5 (y + 2) = 5y + 10, return with the value of 25. When it is substituted with –3, the outcome is that of –5 on both the sides. The fact that this equation may be termed as an identity may also be represented by the factoring the righthand side in such a way that the equation turns out as, 5 (y + 2) = 5 (y + 2) The statements on either side of the equation are the same.
2.3.5. Conditional Equations An expression like 3y + 2 = 11 may be an equality only in the case where y has a specific value. These kinds of statements are known as conditional equations, as they may only hold true when y = 3. In the same way, the equation y + 6 = 13 holds true only when y = 7. The particular value that satisfies a given equation or for which an equation in one variable may hold true is known as the ‘root’ of the equation of the ‘solution’ of it. Thus, solving the algebraic equations generally refers to the case of conditional equations. The solution for an equation, that may be conditional, may be verified by placing the value for the variable that is decided by the root of the equation. The solution is right if the equation comes down to be an identity. For instance, if y is replaced with 3 in the equation 3y + 2 = 11, the outcome is: 3 (3) + 2 = 11 9 + 2 = 11 11 = 11 (an identity) The identity holds true for the value of y = 3, as the value of either side of the equation comes down to 11.
2.3.6. Solving Linear Equations To solve the linear equations in one variable is to find the value for the variable, for which the equation may hold true. For instance, is the solution of the equation x + 9 = 17, as 8 + 9 = 17. The numeric value of 8 is supposed to ‘satisfy’ the equation. Mostly, the methods used to solve the equations are aimed at somehow operating both the sides of the equation by adding,
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subtracting, multiplying, and dividing, till the time when the value of the variable seems obvious. This operation may be done by using the statements in a direct way, which may be concluded in a certain rule which says that if both the sides of the equation are increased, decreased, multiplied or divided by a similar number or an equal value, the outcomes will be equal, except for the case when divided by zero. As indicated previously, an equation may be equated as a balance. The operations done on one side of the equations must be followed by the same operations on the other side, in order to keep a balance. An equation should be kept in balance under all circumstances to ensure that equality is maintained. The rule mentioned above may be used to remove or add terms so as to adjust them in a manner that enables to reach to the value of the variable. These operations may be illustrated in ways given hereafter. •
Addition: Addition may be done in the manner as shown in the equation where the value of x is to be found. The equation is x – 8 = 12 In this equation, the operation of addition may be performed to find the value as: x – 8 + 8 = 12 + 8 x = 20 •
Subtraction: Subtraction may be employed in a way as shown below in the equation,
x + 9 = 23 In this equation, the value may be found as: x + 9–9 = 23–9 x = 14 •
Multiplication: If in the following given equation, the value of y is to be determined,
y = 15 6 Then the number equal to the denominator must be multiplied on both the sides as in
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y × 6 = 15 × 6 6 y = 90 •
Division: If it is instructed to find the value of y in the following equation
6y = 24 the value may be reached at by dividing both the sides or the members by 6, as in
6 y 24 6 = 6 y=4
2.3.7. Operations Required by More Than One Equation Generally, the equations that are required to solve take more steps to reach to their solution than the simple kind of equations that have already been described. However, the basic methods remain the same. If the basic statements are focused very well while solving these equations, the complications will not turn out to be very difficult to handle. Hence, all these equations require is the implementation of one or more of the basic methods of solving the equations put in a logical sequence, to reach to a solution. For instance, if we consider the following equation: 3x + 5 = 23 In this equation, the solution may be reached at by use of two basic operations: subtraction and division. This can be seen as in: Firstly, employing the subtraction method: 3x + 5–5 = 23–5 3x = 18 Secondly, following it up by division method:
3 x 18 3 = 3 x=6
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Another example of the use of three operations together may be taken by solving the following equation:
7y 6 –5=2 Firstly, employing the addition operation:
7y –5+5=2+5 6 7y 6 =7 Secondly, employing the multiplication operation:
7y ×6=7×6 6 7y = 42 Thirdly, making use of the division operation:
7 y 42 = 7 7 y=6 which is the final desired solution.
2.3.8. Equations Having Variable in More Than One Term If an equation is given in the following form and it is asked to find the value of the variable:
3y + y = 26 – y 7 The right member of the equation is made free of the variable by doing the operations as required.
3y + y + y = 26 – y + y 7
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3y + 2y = 26 7 in: 7(
Then, the fraction part is removed by doing the necessary operations as
3y ) + 7 (2y) = 26 × 7 7
3y + 14y = 182 And then normal operations are performed as shown previously: 17y = 182
17 y 182 = 17 17 y=
182 17
which is the required solution.
2.3.9. Quadratic Equations The degree of an equation in one variable is the exponent of the highest power to which the variable is raised in that equation. A seconddegree equation in one variable is one in which the variable is raised to the second power. A seconddegree equation is often called a quadratic equation. The word quadratic is derived from the Latin ward quadratus, which means “squared.” In a quadratic equation, the term of highest degree is the squared term. For example, the following are quadratic equations:
x 2 ℵü x üm + m 2 = The terms of a degree lower than the second may or may not be present. The possible terms of lower degree than the squared term in a quadratic equation are the firstdegree term and the constant term. In the equation 3x2–8x – 5 = 0
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where, –5 is the coefficient of x0. If it is wished to emphasize the powers of x in this equation, the equation could be written in the form 3x2–8x’ – 5x° = 0 Examples of quadratic equations in which either the firstdegree term or the constant term is missing are:
4 x 2 = 16 y 2 + 16 y = 0 e 2 + 12 = 0
2.3.9.1. General Form of a Quadratic Equation Any quadratic equation can be arranged in the general form: ax2 + bx + c = 0 If it has more than three terms, some of them will be alike and can be combined, after which the final form will have at most three terms. For example, 2x2 + 3 + 5x – 1 + x2 = 4 – x2–2x –3 reduces to the simpler form 4x2 + 7x + 1 = 0 In this form, it is easy to see that a, the coefficient of x2 is 4; b, the coefficient of x, is 7; and c, the constant term, is 1. Sometimes the coefficients of the terms of a quadratic appear as negative numbers, as follows: 2x2–3x – 5 = 0 This equation can be rewritten in such a way that the connecting signs are all positive, as in the general form. This is illustrated as follows: 2x2 + (–3) x + (–5) = 0 In this form, the value of a is seen to be 2, b is –3, and c is –5. An equation of the form x +2=0 2
has no x term. This can be considered as a case in which a is 1 (coefficient of x2 understood to be 1), b is 0, and c is 2. For the purpose of emphasizing the values of a, b, and c with reference to the general form, this equation can be written
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x2 + 0x + 2 = 0 The coefficient of x2 can never be 0; If it were 0, the equation would not be a quadratic. If the coefficients of x and x0 are 0, then those terms do not normally appear. To say that the coefficient of x0 is 0 is the same as saying that the constant term is 0 or is missing. A root of an equation in one variable is a value of the variable that satisfied the equation. Every equation in one variable, with constants as coefficients and positive integers as exponents, has as many roots as the exponent of the highest power. In other words, the number of roots is the same as the degree of the equation. A fourthdegree equation has four roots, a cubic (thirddegree) equation has three roots, a quadratic equation has two roots, and a linear equation has one root. As an example, 6 and –1 are roots of the quadratic equation x –5x – 6 = 0 2
This can be verified by substituting these values into the equation and noting that an identity results in each case. Substituting x = 6 gives 62–5(6) – 6 = 0 36–36 = 0 0=0 Substituting x = –1 gives (–1)2–5(–1) – 6 = 0 1 + 5–6 = 0 6–6 = 0 0=0 Several methods of finding the roots of quadratic equations (solving) are possible. The most common methods are solution by factoring and solution by the quadratic formula. Less commonly used methods of solution are accomplished by completing the square and by graphing.
2.3.10. The Discriminant The roots of a quadratic equation may be classified in accordance with the following criteria:
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• Real or imaginary; • Rational or irrational; • Equal or unequal. The task of discriminating among these possible characteristics to find the nature of the roots is best accomplished with the aid of the quadratic formula. The part of the quadratic formula which is used is called the discriminant. If the roots of a quadratic are denoted by the symbols r1 and r2, then the following relations may be stated:
r=
−b ± b 2 − 4ac 2a
where r may be r1 or r2.
We can show that the character of the roots is dependent upon the form taken by the expression b2–4ac which is the quantity under the radical in the formula. This expression is the discriminant of a quadratic equation.
2.3.11. Real and Imaginary Roots Since there is a radical in each root, there is a possibility that the roots could be imaginary. They are imaginary when the number under the radical in the quadratic formula is negative (less than zero). In other words, when the value of the discriminant is less than 0, the roots are imaginary.
x 2 + x + 1 =0 = a 1,= b 1,= c 1 b 2 − 4ac = (1) 2 − 4(1)(1) = 1− 4 = −3 Thus, without further work, we know that the roots are imaginary.
2.3.12. Equal or Double Roots If the discriminant b2–4ac equals zero, the radical in the quadratic formula becomes zero. In this case, the roots are equal; such roots are sometimes called double roots. Consider the equation 9x2 + 12x + 4 = 0
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Comparing with the general quadratic, we notice that a = 9, b = 12, and c = 4 The discriminant is b2–4ac = 122–4 (9) (4) = 144–144 = 0. Therefore, the roots are equal.
2.3.13. Real and Unequal Roots When the discriminant is positive, the roots must be real. Also, they must be unequal since equal roots occur only when the discriminant is zero.
2.4. PROBLEMS TO SOLVE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
A larger integer is 1 more than twiceanother integer. If the sum of the integers is 25, find the integers. If a larger integer is 2 more than 4 times another integer and their difference is 32, find the integers. One integer is 30 more than another integer. If the difference between the larger and twice the smaller is 8, find the integers. The quotient of some number and 4 is 22. Find the number. Eight times a number is decreased by three times the same number, giving a difference of 20. What is the number? One integer is two units less than another. If their sum is −22, find the two integers. The sum of two consecutive integers is 139. Find the integers. The sum of three consecutive integers is 63. Find the integers. The sum of three consecutive integers is 279. Find the integers. The difference of twice the smaller of two consecutive integers and the larger is 39. Find the integers. If the smaller of two consecutive integers is subtracted from two times larger, then the result is 17. Find the integers. The sum of two consecutive even integers is 46. Find the integers. The sum of two consecutive even integers is 238. Find the integers. The sum of three consecutive even integers is 96. Find the integers. If the smaller of two consecutive even integers is subtracted from 3 times larger, the result is 42. Find the integers.
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16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
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The sum of three consecutive even integers is 90. Find the integers. The sum of two consecutive odd integers is 68. Find the integers. The sum of two consecutive odd integers is 180. Find the integers. The sum of three consecutive odd integers is 57. Find the integers. If the smaller of two consecutive odd integers is subtracted from twice larger, the result is 23. Find the integers. Twice the sum of two consecutive odd integers is 32. Find the integers. The difference between twice the larger of two consecutive odd integers and the smaller is 59. Find the integers. Part C: Geometry Problems Set up an algebraic equation and then solve. If the perimeter of a square is 48 inches, then find the length of each side. The length of a rectangle is 2 inches longer than its width. If the perimeter is 36 inches, find the length and width. The length of a rectangle is 2 feet less than twice its width. If the perimeter is 26 feet, find the length and width. The width of a rectangle is 2 centimeters less than onehalf its length. If the perimeter is 56 centimeters, find the length and width. The length of a rectangle is 3 feet less than twice its width. If the perimeter is 54 feet, find the dimensions of the rectangle. If the length of a rectangle is twice as long as the width and its perimeter measures 72 inches, find the dimensions of the rectangle The perimeter of an equilateral triangle measures 63 centimeters. Find the length of each side. An isosceles triangle whose base is onehalf as long as the other two equal sides has a perimeter of 25 centimeters. Find the length of each side. Each of the two equal legs of an isosceles triangle are twice the length of the base. If the perimeter is 105 centimeters, then how long is each leg? A triangle has sides whose measures are consecutive even integers. If the perimeter is 42 inches, find the measure of each side.
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33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.
50.
A triangle has sides whose measures are consecutive odd integers. If the perimeter is 21 inches, find the measure of each side. A triangle has sides whose measures are consecutive integers. If the perimeter is 102 inches, then find the measure of each side. The circumference of a circle measures 50π units. Find the radius. The circumference of a circle measures 10π units. Find the radius. The circumference of a circle measures 100 centimeters. Determine the radius to the nearest tenth. The circumference of a circle measures 20 centimeters. Find the diameter rounded off to the nearest hundredth. The diameter of a circle measures 5 inches. Determine the circumference to the nearest tenth. The diameter of a circle is 13 feet. Calculate the exact value of the circumference. Calculate the simple interest earned on a 2year investment of $1,550 at a 8¾% annual interest rate. Calculate the simple interest earned on a 1year investment of $500 at a 6% annual interest rate. For how many years must $10,000 be invested at an 8½% annual interest rate to yield $4,250 in simple interest? For how many years must $1,000 be invested at a 7.75% annual interest rate to yield $503.75 in simple interest? At what annual interest rate must $2,500 be invested for 3 years in order to yield $412.50 in simple interest? At what annual interest rate must $500 be invested for 2 years in order to yield $93.50 in simple interest? If the simple interest earned for 1 year was $47.25 and the annual rate was 6.3%, what was the principal? If the simple interest earned for 2 years was $369.60 and the annual rate was 5¼%, what was the principal? Joe invested last year’s $2,500 tax return in two different accounts. He put most of the money in a money market account earning 5% simple interest. He invested the rest in a CD earning 8% simple interest. How much did he put in each account if the total interest for the year was $138.50? James invested $1,600 in two accounts. One account earns 4.25% simple interest, and the other earns 8.5%. If the interest
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51.
52.
53.
54.
55. 56. 57. 58. 59. 60. 61. 62. 63.
61
after 1 year was $85, how much did he invest in each account? Jane has her $5,400 savings invested in two accounts. She has part of it in a CD at 3% annual interest and the rest in a savings account that earns 2% annual interest. If the simple interest earned from both accounts is $140 for the year, then how much does she have in each account? Marty put last year’s bonus of $2,400 into two accounts. He invested part in a CD with 2.5% annual interest and the rest in a money market fund with 1.3% annual interest. His total interest for the year was $42.00. How much did he invest in each account? Alice puts money into two accounts, one with 2% annual interest and another with 3% annual interest. She invests three times as much in the higher yielding account as she does in the lower yielding account. If her total interest for the year is $27.50, how much did she invest in each account? Jim invested an inheritance in two separate banks. One bank offered 5.5% annual interest rate and the other 6¼%. He invested twice as much in the higher yielding bank account than he did in the other. If his total simple interest for 1 year was $4,860, then what was the amount of his inheritance? If an item is advertised to cost $29.99 plus 9.25% tax, what is the total cost? If an item is advertised to cost $32.98 plus 8¾% tax, what is the total cost? An item, including an 8.75% tax, cost $46.49. What is the original pretax cost of the item? An item, including a 5.48% tax, cost $17.82. What is the original pretax cost of the item? If a meal costs $32.75, what is the total after adding a 15% tip? How much is a 15% tip on a restaurant bill that totals $33.33? Ray has a handful of dimes and nickels valuing $3.05. He has 5 more dimes than he does nickels. How many of each coin does he have? Jill has 3 fewer halfdollars than she has quarters. The value of all 27 of her coins adds to $9.75. How many of each coin does Jill have? Cathy has to deposit $410 worth of five and tendollar bills. She has 1 fewer than three times as many tens as she does fivedollar bills. How many of each bill does she have to deposit?
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64. 65.
66.
67.
68.
69.
70.
71.
72.
Billy has a pile of quarters, dimes, and nickels that values $3.75. He has 3 more dimes than quarters and 5 more nickels than quarters. How many of each coin does Billy have? Mary has a jar with onedollar bills, halfdollar coins, and quarters valuing $14.00. She has twice as many quarters than she does halfdollar coins and the same amount of halfdollar coins as onedollar bills. How many of each does she have? Chad has a billfold of one, five, and tendollar bills totaling $118. He has 2 more than 3 times as many ones as he does fivedollar bills and 1 fewer ten than fivedollar bills. How many of each bill does Chad have? Part D: Uniform Motion (Distance Problems) Set up an algebraic equation then solve. Two cars leave a location traveling in opposite directions. If one car averages 55 miles per hour and the other averages 65 miles per hour, then how long will it take for them to separate a distance of 300 miles? Two planes leave the airport at the same time traveling in opposite directions. The average speeds for the planes are 450 miles per hour and 395 miles per hour. How long will it take the planes to be a distance of 1,478.75 miles apart? Bill and Ted are racing across the country. Bill leaves 1 hour earlier than Ted and travels at an average rate of 60 miles per hour. If Ted intends to catch up at a rate of 70 miles per hour, then how long will it take? Two brothers leave from the same location, one in a car and the other on a bicycle, to meet up at their grandmother’s house for dinner. If one brother averages 30 miles per hour in the car and the other averages 12 miles per hour on the bicycle, then it takes the brother on the bicycle 1 hour less than 3 times as long as the other in the car. How long does it take each of them to make the trip? A commercial airline pilot flew at an average speed of 350 miles per hour before being informed that his destination airfield may be closed due to poor weather conditions. In an attempt to arrive before the storm, he increased his speed 400 miles per hour and flew for another 3 hours. If the total distance flown was 2,950 miles, then how long did the trip take? Two brothers drove the 2,793 miles from Los Angeles to New York. One of the brothers, driving during the day, was able to
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74. 75.
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average 70 miles per hour, and the other, driving at night, was able to average 53 miles per hour. If the brother driving at night drove 3 hours less than the brother driving in the day, then how many hours did they each drive? Joe and Ellen live 21 miles apart. Departing at the same time, they cycle toward each other. If Joe averages 8 miles per hour and Ellen averages 6 miles per hour, how long will it take them to meet? If it takes 6 minutes to drive to the automobile repair shop at an average speed of 30 miles per hour, then how long will it take to walk back at an average rate of 4 miles per hour? Jaime and Alex leave the same location and travel in opposite directions. Traffic conditions enabled Alex to average 14 miles per hour faster than Jaime. After 1½ hours they are 159 miles apart. Find the speed at which each was able to travel. Jane and Holly live 51 miles apart and leave at the same time traveling toward each other to meet for lunch. Jane traveled on the freeway at twice the average speed as Holly. They were able to meet in a half hour. At what rate did each travel?
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REFERENCES 1.
2.
Elementary Algebra, (n.d.). [eBook] Available at: https://www.saylor. org/site/textbooks/Elementary%20Algebra.pdf [Accessed 24 August 2018]. Mathematics, Basic Math and Algebra, (2003). [eBook] Naval Education and Training Professional Development and Technology Center. Available at: http://www.cbtricks.com/miscellaneous/tech_ publications/neets/basic_math_and_algebra.pdf [Accessed 24 August 2018].
3 Number Theory and Number System
CONTENTS 3.1. Introduction ...................................................................................... 66 3.2. Number Theory................................................................................. 68 3.3. Facts About Number Theory .............................................................. 70 3.4. Number Systems, Base Conversions, And Computer Data Representation ....................................................................... 71 3.5. Conversions ...................................................................................... 78 3.6. Number Systems ............................................................................... 80 3.7. Conclusion ....................................................................................... 84 References ............................................................................................... 85
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This chapter starts with the history of how the numbers were invented and developed and are followed by the introduction to the Number Theory. We will also get into the details of why the number theory is important in mathematics, and also, how we use them in our daytoday life. Number theory also deals with the natural numbers that are used by everyone – directly or indirectly. This chapter also explains the different types of numbers along with their examples and the concepts – how they can be implemented in getting a solution to a problem. Added to it, we have also mentioned some amazing facts about the uses of Number Theory.
3.1. INTRODUCTION 3.1.1. History of Numbers The ability to count dates back to prehistoric times. This is evident from archaeological artifacts, such as a 15,000yearold bone from the Congo region of Africa with tally marks scratched upon it—signs of an unknown ancestor counting something. Very near the dawn of civilization, people had grasped the idea of “multiplicity” and thereby had taken the first steps toward a study of numbers. It is certain that an understanding of numbers existed in ancient Mesopotamia, India, China, and Egypt for tablets, papyri, and temple carvings from these early cultures have survived. Indian mathematicians were hard at work. In the 7th century Brahmagupta took up what is now (erroneously) called the Pell equation. He posed the challenge to find a perfect square that, when multiplied by 92 and increased by 1, yields another perfect square. That is, he sought whole numbers x and y such that 92x2 + 1 = y2—a Diophantine equation with quadratic terms. Brahmagupta suggested that anyone who could solve this problem within a year earned the right to be called a mathematician. His solution was x = 120 and y = 1.151. In addition, Indian scholars developed the socalled HinduArabic numerals—the base–10 notation subsequently adopted by the world’s mathematical and civil communities. Although there were number representation than number theory, these numerals have prevailed due to their simplicity and ease of use. The Indians employed this system— including the zero—as early as AD900.
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At about this time, the Islamic world became a mathematical powerhouse. Situated on trade routes between East and West, Islamic scholars absorbed the works of other civilizations and augmented these with homegrown achievements.
3.1.2. Introduction to Number Theory Number theory is a branch of mathematics and is concerned with the properties of the positive integers (1, 2, 3, …). Sometimes called “higher arithmetic,” it is among the oldest and most natural method of mathematical pursuits. Number theory has always fascinated amateurs as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background. Until the mid20th century, number theory was considered the purest branch of mathematics, with no direct applications to the real world. With the advent of computers and digital communications revealed that number theory could provide unanticipated answers to realworld problems. At the same time, improvements in computer technology enabled number theorists to make remarkable advances in factoring large numbers, determining primes numbers, testing conjectures, and solving numerical problems once considered out of reach. Modern number theory is a broad subject that is classified such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning to the integers.
3.1.3. Why is Number Theory Important Number Theory’s importance is in that it has forever led the way in the world of mathematics. If it weren’t for number theory we wouldn’t know nearly as much as we know about groups, rings, fields, Galois theory, Algebraic Geometry (the Italian college, prior to Zariski and Grothendieck’s time, was far behind relative to our current understanding of Algebraic Geometry— and it’s all due to the ideas seeping in from Number Theory. There are even some recent papers implying that perhaps the correct way to view Quantum Field Theory is through a geometric analog of the Langlands Program—an extremely complicated theory in Number Theory.
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3.2. NUMBER THEORY “Number theory is a branch of pure mathematics devoted to the study of the natural numbers and the integers. Number Theory is partly experimental and partly theoretical.” Number theory is the study of the set of positive whole numbers 1, 2, 3, 4, 5, 6, 7, which are often called the set of natural numbers. The relationships between different sorts of Natural Numbers have been separated by people in ancient times. The different varieties of Natural Numbers are: • Odd – 1, 3, 5, 7, 9, 11. • Even – 2, 4, 6, 8, 10. • Square – 1, 4, 9, 16, 25, 36. • Cube – 1, 8, 27, 64, 125. • Prime – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31. • Composite – 4, 6, 8, 9, 10, 12, 14, 15, 16. • 1 (modulo 4) – 1, 5, 9, 13, 17, 21, 25. • 3 (modulo 4) – 3, 7, 11, 15, 19, 23, 27. • Triangular – 1, 3, 6, 10, 15, 21. • Perfect – 6, 28, 496. • Fibonacci – 1, 1, 2, 3, 5, 8, 13, 21. Many of these types of numbers are undoubtedly already known to us other than the Modulo 4 which may not be quite familiar. Now let us see the different varieties of Natural Numbers in detail.
3.2.1. Odd Numbers Odd numbers cannot be divided evenly into groups of two. The number five can be divided into two groups of two and one group of one. Odd numbers always end with a digit of 1, 3, 5, 7, or 9. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31 are odd numbers.
3.2.2. Even Numbers Even numbers can be divided evenly into groups of two. The number four can be divided into two groups of two.
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Even numbers always end with a digit of 0, 2, 4, 6 or 8. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 are even numbers.
3.2.3. Square Numbers A square number is a number multiplied by itself. This can also be called ‘a number squared.’ The symbol for squared is ². 2² = 2 x 2 = 4 3² = 3 x 3 = 9 4² = 4 x 4 = 16 5² = 5 x 5 = 25 The square numbers up to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
3.2.4. Cube Numbers A cube number is a number multiplied by itself 3 times. This can also be called ‘a number cubed.’ The symbol for cubed is ³. 2³ = 2 × 2 × 2 = 8 3³ = 3 × 3 × 3 = 27 4³ = 4 × 4 × 4 = 64 5³ = 5 × 5 × 5 = 125 The cube numbers up to 100 are: 1, 8, 27, 64.
3.2.5. Prime Numbers A prime number is a whole number that is greater than one and has exactly two factors, 1 and itself. Number 2 is the only prime even number. The prime numbers less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
3.2.6. Composite Numbers A composite number is a whole number that is greater than one and has more than two factors. The number 1 is neither prime nor composite. (modulo 4) and 3 (modulo 4): A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and similarly for the 3 (modulo 4) numbers.
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3.2.7. Triangular A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on.
3.2.8. Perfect Number A number is perfect if the sum of all its divisors, other than itself, adds back up to the original number.
3.2.9. Fibonacci The Fibonacci numbers are created by starting with 1 and 1. Then, to get the next number in the list, just add the previous two. Finally, the numbers dividing 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Similarly, the divisors of 28 are 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28.
3.3. FACTS ABOUT NUMBER THEORY • •
• •
•
•
All four digit palindromic numbers are divisible by 11. If we repeat a threedigit number twice, to form a sixdigit number. The result will be divisible by 7, 11 and 13, and dividing by all three will give your original threedigit number. A number of form 2N has exactly N + 1 divisors. For example: 4 has 3 divisors, 1, 2 and 4. To calculate the sum of factors of a number, we can find the number of prime factors and their exponents. Let p1, p2, … pk be prime factors of n. Let a1, a2. ak be highest powers of p1, p2. pk respectively that divide n, i.e., we can write n as n = (p1a1)*(p2a2)* … (pkak). For a product of N numbers, if we have to subtract a constant K such that the product gets its maximum value, then subtract it from the largest value such that largest valuek is greater than 0. If we have to subtract a constant K such that the product gets its minimum value, then subtract it from the smallest value where the smallest valuek should be greater than 0 Goldbach’s conjecture: Every even integer greater than 2 can be expressed as the sum of 2 primes.
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•
•
•
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Perfect numbers or Amicable numbers: Perfect numbers are those numbers which are equal to the sum of their proper divisors. Example: 6 = 1 + 2 + 3. Lychrel numbers: Are those numbers that cannot form a palindrome when repeatedly reversed and added to itself. For example: 47 is not a Lychrel Number as 47 + 74 = 121. Lemoine’s Conjecture: Any odd integer greater than 5 can be expressed as a sum of an odd prime (all primes other than 2 are odd) and an even semiprime. A semiprime number is a product of two prime numbers. This is called Lemoine’s conjecture. Fermat’s Last Theorem: According to the theorem, no three positive integers a, b, c satisfy the equation, for any integer value of n greater than 2. For n = 1 and n = 2, the equation have infinitely many solutions.
3.4. NUMBER SYSTEMS, BASE CONVERSIONS, AND COMPUTER DATA REPRESENTATION 3.4.1. Decimal and Binary Numbers To write a decimal number (base 10), a positional notation system is used. On the basis of the position of a number, each digit is multiplied by an appropriate power of 10. The same can be described as given below: 749 = 7 × 102 + 4 × 101 + 9 × 100 = 7 × 100 + 4 × 10 + 9 × 1 = 700 + 40 + 9 In the case of whole numbers, the rightmost digit position is the one’s position (100 = 1). Any number at that position would indicate the total number of one’s presence in the number. The next position to the left is ten’s, then comes the hundred’s position, thousand’s, and so on. Each digit position has a weight that is ten times the weight of the position to its right. In the decimal number system, there is a possibility of ten values appearing in each digit position. Therefore, there are ten numerals which are supposed to be represented for the quantity in each digit position. The decimal numerals are the familiar zero through nine (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9). In a positional notation system, the number base is
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popularly known as the radix. Thus, the base 10 system, which is mostly used comprises a radix of 10. The term radix and base are almost similar and can be used interchangeably. When writing numbers in a radix other than ten, or where the radix isn’t clear from the context, it is customary to specify the radix using a subscript. Thus, in a case where the radix isn’t understood, decimal numbers would be written like this: 26710 2910672110
Generally, the radix will be understood from the context and the radix specification is left off. The binary number system is also a positional notation numbering system, but here the base is not exactly 10, but it is termed and recognized as two. This is so because each digit position in a binary number represents a power of two. So, when a binary number is written, each binary digit is multiplied by an appropriate power of 2 based on the position in the number: For example: 101101 = 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 = 1× 32 + 0 × 16 + 1× 8 + 1 × 4 + 0 × 2 + 1 × 1 = 32 + 8 + 4 + 1
In the binary number system, there are only two possible values that can appear in each digit position rather than the ten that can appear in a decimal number. Only the numerals 0 and 1 are used in binary numbers. The term ‘bit’ is a contraction of the words ‘binary’ and ‘digit,’ and when it comes to binary numbers the terms bit and digit can also be used interchangeably. When talking about binary numbers, it is generally required to have an overview of those number of bits used to store or represent the number. This merely describes the number of binary digits that would be required to write the number. The number in the above example is a 6bit number. The following are some additional examples of binary numbers: 1011012 112 101102
3.4.2. Hexadecimal Numbers In addition to binary, there is another set of number base that is commonly used in digital systems, and that is base 16. This number system is known as hexadecimal, and each digit position represents a power of 16. For any number base greater than ten, a problem occurs because there are more than ten symbols needed to represent the numerals for that number base. It is customary in these cases to use the ten decimal numerals followed by the
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letters of the alphabet beginning with A to provide the needed numerals. Since the hexadecimal system is base 16, there are sixteen numerals required. The following are the hexadecimal numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F The following are some examples of hexadecimal numbers: 1016
4716
3FA16 A03F16
The reason for the common use of hexadecimal numbers is the relationship between the numbers 2 and 16. Sixteen is a power of 2 (16 = 24). Because of this relationship, four digits in a binary number can be represented with a single hexadecimal digit. This makes conversion between binary and hexadecimal numbers very easy, and hexadecimal can be used to write large binary numbers with much fewer digits. When working with large digital systems, such as computers, it is common to find binary numbers with 8, 16 and even 32 digits. Writing a 16 or 32bit binary number would be quite tedious and errorprone. By using hexadecimal, the numbers can be written with fewer digits and much less likelihood of error. To convert a binary number to hexadecimal, divide it into groups of four digits starting with the rightmost digit. If the number of digits isn’t a multiple of 4, prefix the number with 0’s so that each group contains 4 digits. For each fourdigit group, convert the 4bit binary number into an equivalent hexadecimal digit. (See the Binary, BCD, and Hexadecimal Number Tables at the end of this document for the correspondence between 4bit binary patterns and hexadecimal digits). For example: Convert the binary number 10110101 to a hexadecimal number
Another example: Convert the binary number 0110101110001100 to hexadecimal
In order to convert a hexadecimal number into a binary number, it is first required to convert each hexadecimal digit into a group of 4 binary digits.
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Example: let say, convert the hex number 374F into binary a binary format, then it could be represented as:
To prove that the abovementioned representation is a hexadecimal representation, there are various techniques that can be used for the same instead of using some other radix. In cases where the context makes it absolutely clear that numbers are represented in hexadecimal, there are no requirements of an indicator. In muchwritten material where the context doesn’t make it clear what the radix is, the numeric subscript 16 following the hexadecimal number is used. In most of the programming languages, this method is not very productive. Therefore, there are few conventions that would be used, and these conventions are dependent upon the programming language. In the C and C + + languages, hexadecimal constants are represented with a ‘0x’ preceding the number, as in: 0x317F, or 0x1234, or 0xAF. In assembler programming languages that follow the Intel style, a hexadecimal constant begins with a numeric character (so that the assembler can distinguish it from a variable name), a leading ‘0’ being used if necessary. The letter ‘h’ is then suffixed onto the number to inform the assembler that it is a hexadecimal constant. In Intel style assembler format: 371Fh and 0FABCh are valid hexadecimal constants. The programmer should also notice that, A37h isn’t a valid hexadecimal constant. It doesn’t begin with a numeric character, and so will be taken by the assembler as a variable name. In assembler programming languages that follow the Motorola style, hexadecimal constants begin with a ‘$’ character. So, in this case: $371F or $FABC or $01 are valid hexadecimal constants.
3.4.3. Binary Coded Decimal Numbers Number systems are of various kinds. One of the most commonly used number systems is the Binary Coded Decimal number system which is used occasionally. In this system, numbers are represented in a decimal form, wherein each decimal digit is encoded using a fourbit binary number. For example, the decimal number 136 would be represented in Binary Coded Decimal as:
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0001 0011 0110 1 3 6
The conversion of numbers between decimal and Binary Coded Decimal is not a difficult task to perform. Under the conversion process of a number from decimal to Binary Coded Decimal, first, it is required to write down the fourbit binary pattern for each decimal digit. The second step would be to convert the digits from Binary Coded Decimal to decimal, where it is required that the numbers would be divided into groups of 4 bits and a decimal digit would be written corresponding to each 4digit group. There are a couple of variations on the Binary Coded Decimal representation, and those variations are known as namely packed and unpacked variations. An unpacked Binary Coded Decimal number has only a single decimal digit stored in each data byte. In this case, the decimal digit will be in the low 4 bits, and the upper 4 bits of the byte will be 0. In the case of the packed Binary Coded Decimal representation, two decimal digits are placed in each byte. Generally, the high order bits of the data byte contain the more significant decimal digit.
The Binary Coded Decimal is not very commonly used when compared with the other number systems, especially in the computer system and programming. This is so because this system is not very space efficient. In packed Binary Coded Decimal, only 10 of the 16 possible bit patterns in each 4bit unit are used. In unpacked Binary Coded Decimal, only 10 of the 256 possible bit patterns in each byte are used. A 16bit quantity can represent the range 0–65535 in binary, 0–9999 in packed Binary Coded Decimal and only 0–99 in unpacked Binary Coded Decimal.
3.4.4. Fixed Precision and Overflow Binary number system can also be used in the case of the maximum size of the number. Generally, in most of the computer systems, it is not necessary
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to have a binary system for all the numbers. Numbers in computers are typically represented using a fixed number of bits. These sizes are typically 8 bits, 16 bits, 32 bits, 64 bits, and 80 bits. All these sizes are generally a multiple of 8, as most computer memories are organized on an 8bit byte basis. Numbers in which a specific number of bits are used to represent the value, then such numbers are known as fixed precision numbers. In the case of representing a number with a specific number of bits, then this signifies the range of possible values that can be represented. For example, there are 184 possible combinations of 8 bits; therefore an 8bit number can represent 184 distinct numeric values, and the range is typically considered to be 0–183. Any number larger than 183 would not be represented using 8 bits. Similarly, 16 bits allows a range of 0–65535. When fixed precision numbers are used, (as they are in virtually all computer calculations) then the programmer must take into consideration the concept of overflow. An overflow could occur in that situation where the result of a calculation is not represented with the number of bits available. For example, when adding the two eightbit quantities: 270 + 380, the result is 650. This is not in the range of 0–255, and so the result can’t be represented using 8 bits. This signifies that the result has overflowed the available range. When overflow takes place, the low order bits of the result will be valid, but the high order bits will be lost. This results in a value that is significantly smaller than the correct result. When doing fixed precision arithmetic (which all computer arithmetic involves), it is necessary to be conscious of the possibility of overflow in the calculations.
3.4.5. Signed and Unsigned Numbers Until now, only positive values for binary numbers were taken into consideration. In the case of a fixed precision binary number where the positive values are kept on hold, where the binary number is said to be unsigned. Under such special precision cases, there is a specific range of positive values which generally tends to start from 0, …., 2n –1, where n is the number of bits used. But, apart from using only positive values, negative values can also be taken into consideration in binary. In this case, part of the total range of values is used to represent positive values, and the rest of the range is used to represent negative values.
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There are various techniques that would help in representing sign values in binary. And, the most common technique used for representing the signed values in binary is known as the “two’s complement.” The term “two’s complement” is somewhat ambiguous, as it can be used in two different ways. Those two ways are: •
First, as a representation, two’s complement is a way of interpreting and assigning meaning to a bit pattern contained in a fixed precision binary quantity; and • Second, the term two’s complement is also used to refer to an operation that can be performed on the bits of a binary quantity. As an operation, the two’s complement of a number is formed by inverting all of the bits and adding 1. In a binary number being interpreted using the two’s complement representation, the high order bit of the number indicates the sign. If the sign bit is 0, the number is positive, and if the sign bit is 1, the number is negative. For positive numbers, the rest of the bits hold the true magnitude of the number. For negative numbers, the lower order bits hold the complement (or bitwise inverse) of the magnitude of the number. It is important to note that two’s complement representation can only be applied to fixed precision quantities, that is, quantities where there are a set number of bits. Two’s complement representation is used because it reduces the complexity of the hardware in the arithmeticlogic unit of a computer’s CPU. Using a two’s complement representation, all of the arithmetic operations can be performed by the same hardware whether the numbers are considered to be unsigned or signed. The bit operations performed are identical, the difference comes from the interpretation of the bits. The interpretation of the value will be different depending on whether the value is considered to be unsigned or signed.
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If noticed rightly in the above sequence, that the counting up from 0, and when it reaches is to 127, the next binary pattern in the sequence corresponds to –128. This shows that the values jump from the greatest positive number to the greatest negative number. But, such situations are expected after a certain point of time, i.e., adding 1 to –128 yields –127, and so on. When the count has progressed to 0FFh (or the largest unsigned magnitude possible), the count wraps around to 0. (i.e., adding 1 to –1 yield 0).
3.5. CONVERSIONS 3.5.1. Binary Number to Decimal Number To convert a binary number to decimal number is an easy task. All that one has to do is to find the decimal value of each binary digit position that contains a 1 and perform the addition operator on it. For example: convert 101102 to decimal. 10110 \ \ \___________1 x 21 = 2 \ \____________1 x 22 = 4 \_______________1 x 24 = 16
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The decimal number for the above given binary number is 22 Another example: convert 110112 to decimal 11011 \\\
\_________1 x 20 = 1
\\
\__________1 x 21 = 2
\ \_____________1 x 23 = 8 \______________1 x 24 = 16 The decimal number for the above given binary number is 27
3.5.2. Decimal Number to Binary Number The procedure by which we convert a decimal number to a binary number is the one that can be used to perform the conversions of a decimal number to any number base. The method is the successive division by the radix until at one point, and the dividend reaches 0. At each stage of division, the remainder provides a digit of the converted number, which starts with the least significant digit. Taking an example to understand the conversion: convert 3710 to binary 37/2 = 18
remainder 1
18/2 = 9
remainder 0
9/2 = 4
remainder 1
4/2 = 2
remainder 0
2/2 = 1
remainder 0
1/2 = 0
remainder 1
(least significant digit)
(most significant digit)
The resulting binary number is: 100101 Another example: convert 9310 to binary 93/2 = 46 remainder 1 (least significant digit) 46/2 = 23 remainder 0 23/2 = 11 remainder 1 11/2 = 5 remainder 1 5/2 = 2 remainder 1 2/2 = 1 remainder 0 1/2 = 0 remainder 1 (most significant digit)
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3.6. NUMBER SYSTEMS The number system is a basic topic when it comes to the fundamentals of advanced mathematics. In this chapter, we will discuss the number in greater depth that will include the binary number system, integers, the real and complex numbers and the rational.
3.6.1. Binary Number System Binary number system means numbers with the base as 2 (the prefix bi). On the basis of topics earlier discussed, the decimal will correlate the previous knowledge of the decimal number system to the binary number system, the digits used to count in this number system are 0 and 1. This will lay the foundations on which the discussion will vary, and the numbers used, i.e., 1 and 0 are called the binary digits (bits), representation schemes for numbers, that is, the real numbers and integers, will be leased.
3.6.2. General Properties in Arithmetic Let R be a set and let. And + be the binary operators that will be used on R (called the product and sum, respectively), so that x + y ∈ R and xy ∈ R whenever x, y ∈ R. Further, (R., + ) is called a ring, in case, the following conditions hold for all a, b, c ∈ R: A1: Commutative law for + a+b=b+a
A2: Associative law for + a + (b + c) = (a + b) + c A3: There exists 0 ∈ R, such that 0 + a = a for all a ∈ R (zero)
A4: For every a ∈ R there exists −a ∈ R such that a + (−a) = 0 (negatives) M1: Commutative law for . ab = ba
M2: Associative law for . a (bc) = (ab)c M3: There exists 1 ∈ R such that 1≠0 and 1a = a for all a ∈ R (unity)
D: Distributive law a (b + c) = ab + ac
(R, + , ·) is known as a field if, in addition,
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M4: For every a ∈ R with a≠ 0 there exists a−1∈ R such that aa−1 = 1 (inverses).
[Jumping forward, Z is a ring and Q, R, C are fields.]
Properties: By using these properties, we can show that in any ring:
• 0 is unique; • 1 is unique; • for all a ∈ R, −(−a) = a; • 0a = a0 = 0 for all a ∈ R. Sample proofs: (ii) Let us suppose 1 and q both mollify (M3). Then q = 1q = q1 = 1 using (M3), (M1) and that q mollifies (M3). Note that a0 + a0 = a (0 + 0) = a0 by (D) and (A3). Therefore, a0 = a0 + 0 = a0 + (a0 + −(a0)) = (a0 + a0) + −(a0) = a0 + −(a0) = 0, using (A3), (A1), (A4), (A2) and (A4). Cancellation laws: The general cancellation laws also follow from the field or ring axioms. C1: In any ring b + a = c + a implies that b = c C2: In any field ba = ca and a ≠ 0 implies b = c. Proof: If b + a = c + a then, b + (a + −a) = (b + a) + −a = (c + a) + −a = c + (a + −a) using (A2), so by (A4), (A3), and (A1) b = b + 0 = c + 0 = c. (C2) is similar. Order Properties: We will also be concerned with the notion of ‘order’ on a ring. A ring ‘R’ is ordered if there is an order relation ≤ on R such that for all a, b, c, d ∈ R we have either a ≤ b or b ≤ a with both holding if and only if a = b and: • O1: a ≤ b, b ≤ c implies a ≤ c; • O2: a ≤ b, c ≤ d implies a + c ≤ b + d; • O3: 0 ≤ a ≤ b, 0 ≤ c ≤ d implies ac ≤ bd. Obviously, once we have described ≤ then the definitions of ≥, < and > follow by the general conventions. e.g., a < b means a ≤ b and a ≠ b.
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3.6.3. Integers One has to begin from anywhere and in this course, we will begin from the natural numbers and construct up other number systems from there. It is very possible to take a further source starting point, namely the Peano Axioms, and advance the natural numbers and the integers from there. Therefore, we will accept that the integers Z = {.−3, −2, −1, 0, 1, 2, 3.} are ‘familiar’ and ‘wellunderstood.’ In specific, we assume the typical properties of addition, multiplication, and order. Thus: Proposition: The integers Z form a ring under normal addition and multiplication, and this ring is ordered under ≤. The integers are suitable for various things, in specific for purposes of mathematical induction. Though, they are in some ways inadequate, particularly that integers do not have multiplicative inverses, in other words, that they do not form a field.
3.6.4. Rational Numbers Why do we need to define rational numbers? Why are we not content with our intuition of them (as we pretend to be with Z)? Well, we just approximately could. The intangible difficulty arising with rational numbers is that different forms, for example, say, 3/5 and 6/10, essentially represent the same rational number. We will observe how the language of equivalence relations can be utilized to overcome this issue. But this will be a warmup exercise for the building up of real numbers from the rational – where our perception is much flakier. A rational number for example 3/5, is just an elaborate notation for an ordered pair of integers (3, 5) and we require to identify this with the pairs (6, 10), (9, 15), etc., which are a distinct representation of the same fraction. Write Z∗ = Z\ 0 for the set of nonzero integers. We define a relation on Z×Z∗ by (a, b) R (c, d) ⇐⇒ ad = bc.
Theorem: R is an equivalence relation. Proof: By using the properties of multiplication of integers:
R: ab = ba ⇒ (a, b) R (a, b).
S: (a, b) R (c, d) ⇒ ad = bc ⇒ cb = da ⇒ (c, d) R (a, b). T: (a, b) R (c, d) & (c, d) R (e, f) ⇒ ad = bc & c f = de
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⇒ ad f = bc f = bde ⇒ a f = be (as d 6 = 0) ⇒ (a, b) R (e, f).
Definition: We call the equivalence classes of R the rational numbers: Q = {[(x, y)]: (x, y) ∈ X}.
The equivalence class [(a, b)] is denoted by a/b. Example: Define the equivalence classes of (0, 1) and (1, 1). We require to define the elementary operations on Q. We do this by ‘mimicking’ what we ‘already know’: (a/b) + (c/d) = (ad + bc)/bd [ (a, b)] + [(c, d)] = [(ad + bc, bd)] (a/b).(c/d) = ac/bd [(a, b)][(c, d)] = [(ac, bd)] Along with the modular arithmetic, we are describing operations for equivalence classes by using their representatives. We require to check that: Proposition: The operations of ‘ + ’ and ‘·’ on Q given above are welldefined. Proof: We will prove this for addition only. Let us suppose that, [(a1, b1)] = [(a2, b2)], [(c1, d1)] = [(c2, d2)]. (3.1) Then, a1b2 = b1a2, c1d2 = d1c2. Using the ring properties of + and · on Z, we have, (a1d1 + b1c1) b2d2 = a1d1b2d2 + b1c1b2d2 = a1b2d1d2 + b1b2c1d2 = b1a2d1d2 + b1b2d1c2 = b1d1(a2d2 + b2c2). This basically means that, (a1d1 + b1c1, b1d1) R (a2d2 + b2c2, b2d2), and therefore, [(a1d1 + b1c1, b1d1)] = [(a2d2 + b2c2, b2d2)], as required.
3.6.5. Complex Numbers By definition, the product of cuts we see that x2 ≥ 0 for all x ∈ R, so it follows that negative numbers do not have square roots or, certainly, any even roots. In specific, there is no number x ∈ R satisfying x2 = −1.
We can ‘invent’ such a number, and ‘adjoin’ it to R. This is how we construct complex numbers. But then approximately quite notable happens: it results that all real numbers have all roots. Even better, n√z exists for all
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the complex number z and allnatural number n that is all complex numbers have all roots. Even improved than that, each polynomial equation P(z) = 0, where P is a polynomial with complex coefficients, has a solution in complex numbers. This latter result is known as the Fundamental Theorem of Algebra. As compared to the build ups of reals from rational, or even the builds up of rational from integers, the buildup of complex numbers from reals is simple and direct. Definition: C = R×R = {(x, y): x, y ∈ R}.
In other words, complex numbers are defined as the ordered pairs of reals. Addition and multiplication are defined as follows: (a, b) + (c, d) = (a + c, b + d), (a, b)·(c, d) = (ac−bd, ad + bc).
3.7. CONCLUSION There are many open questions as well as the directions for further research. It can be noted that all have a certain “direction” of evolution. Thus, we can come to the conclusion that the Number Theory and Number System is the “queen of mathematics” as it has helped our mathematicians to develop and advance the various departments in mathematics.
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REFERENCES 1.
Aaamath.com., (2012). Even and Odd Numbers. [online] Available at: http://www.aaamath.com/nam25ax2.htm [Accessed 24 August 2018]. 2. Bbc.com., (2018). What are Square and Cube Numbers? [online] Available at: https://www.bbc.com/bitesize/articles/z2ndsrd [Accessed 24 August 2018]. 3. Chapter 1 What is Number Theory? (n.d.). [eBook] Available at: https://www.math.brown.edu/~jhs/frintch1ch6.pdf [Accessed 24 August 2018]. 4. Chapter 10: Number Systems and Arithmetic Operations, (n.d.). [eBook] Available at: http://www.pbte.edu.pk/text%20books/dae/ math_123/Chapter_10.pdf [Accessed 23 August 2018]. 5. Encyclopedia Britannica, (2018). Number Theory – Euclid. [online] Available at: https://www.britannica.com/science/numbertheory/ Euclid [Accessed 24 August 2018]. 6. Falconer, K., (2011). Fundamentals of Pure Mathematics. [eBook] Available at: http://wwwmaths.mcs.standrews.ac.uk/~kenneth/ FundPureNotes.pdf [Accessed 24 August 2018]. 7. Geeks for geeks, (2018). Number Theory (Interesting Facts and Algorithms) – Geeks for Geeks. [online] Available at: https://www. geeksforgeeks.org/numbertheoryinterestingfactsandalgorithms/ [Accessed 24 August 2018]. 8. Mi.sanu.ac.rs., (n.d.). Conclusion. [online] Available at: https://www. mi.sanu.ac.rs/vismath/kocic/ch4.htm [Accessed 24 August 2018]. 9. Number Systems, Base Conversions, and Computer Data Representation, (n.d.). [eBook] Available at: https://www.eecs.wsu. edu/~ee314/handouts/numsys.pdf [Accessed 23 August 2018]. 10. Ramagge, J., Brown, P., Evans, M., Hunt, D., McIntosh, J., & Pender, B., (2011). The Real Numbers. [eBook] Available at: http://www.amsi. org.au/teacher_modules/pdfs/Real_numbers.pdf [Accessed 24 August 2018]. 11. The Economic Times, (2018). Definition of Number Theory  What is Number Theory? Number Theory Meaning – The Economic Times. [online] Available at: https://economictimes.indiatimes.com/definition/ numbertheory [Accessed 24 August 2018].
4 Relations and Functions
CONTENTS 4.1. Introduction ...................................................................................... 88 4.2. Binary Relations................................................................................ 94 4.3. Functions .......................................................................................... 98 4.4. Case Study: A MathematicalAlgorithmic Approach To Sets............. 110 References ............................................................................................. 114
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This chapter gives an introduction to the relations and functions which are often considered as fundamental building blocks in mathematics. Relations point out the relationships between members of two sets A and B and functions are defined as a special kind of relation where there is accurately or at most one relationship for each element a ∈ A with an element in B. A set is defined as a collection of welldefined objects which contains no duplicates. The term “well defined” describes that for a given value it is likely to determine whether or not it is a member of the set. Functions can be total or partial. A total function f: A→B is a special relation such that for each element a ∈ A there is precisely one element b ∈ B. This is written as f (a) = b. A partial function differs from a total function in that the function may be undefined for one or more values of A. The domain of a function which is denoted by dom f is the set of values in A for which the function is defined. The domain of the function is A provided that f is a total function. The codomain of the function is B. This chapter will further define all the terminologies and concept of relations and function along with the examples.
4.1. INTRODUCTION 4.1.1. Ordered Pairs and Cartesian Products As we observe, there is no order forced on the elements of a set. To define relations and functions, we will require the concept of an ordered pair, for example, in which ‘a’ is considered as the first element and ‘b’ is the second element of the pair. So, in common, ≠ < b, a >. (Whereas for a set, {a, b} = {b, a}.). Many mathematicians think that there is no way to describe ordered pair in context of sets as sets themselves are unordered. There are various ways by which this can be defined. The most conventional way to define the ordered pair in sets is as follows: Definition: = def {{a}, {a, b}} The critical part is that for each ordered pair, there is definitely exactly one equivalent set of the form {{a}, {a, b}}, and two different ordered pairs always have two different equivalent sets. There would not be anything wrong with captivating the concept of ordered pair as another primitive notion, in conjunction with the notion of set.
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But different mathematicians like considering how far they can diminish the number of primitives, and it’s an exciting discovery to see that the notion of order can be described in terms of set theory. Cartesian product: Let us suppose we have two sets A and B. We make ordered pairs by taking an element of A as the first member of the pair and an element of B as the second member. The Cartesian product of A and B, written as A × B, is the set consisting of all such pairs. The predicate information describes it as: A × B = def {  x ∈ A and y ∈ B}
The description of ordered pairs can be extensive to ordered triples and in broadspectrum to ordered ntuples for any natural n. For example, ordered triples are basically described as: = def The Cartesian product for three sets A, B, and C can be defined as (Figure 4.1): A × B × C = def ((A × B) × C)
Figure 4.1: Cartesian product. Source: https://www.askiitians.com/iitjeealgebra/setrelationsfunctions/cartesianproduct.aspx
4.1.2. Relations Naturally, a relation is the kind of thing which either does or does not grip between certain things, for example the love relation holds between Kelly and John just in case Kelly loves John, and the lessthan relation grasps between two natural numbers A and B just in case A < B.
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A simpleminded first pass might be to represent the love relation as the set of all pairs {A, B} such that A and B are two people such that A loves B. Essentially, A, and B would not be people at all, but rather certain sets that we have chosen as theoretical representations of people. It must be noted that the only things in our mathematical workspace are sets! Inappropriately, this is very simple, as, for example, we are left with no method to represent unreciprocated love: what if Kelly loves John but John does not love Kelly? A more auspicious approach is to represent love as the set of ordered pairs (A, B) such that A loves B. Certainly, nobody is below the illusion that a set of ordered pairs is the response Cole Porter had in mind when he wrote: What is this Thing Called Love? It is what a formal semanticist would call the continuation of the love relation. The suitable way to represent mathematically the real love relation, as opposed to its continuation, is a question we will turn to later when we consider how to represent linguistic meaning. To take a less worrying example, we can ponder the relation ⊆U of set inclusion constrained to the subsets of a given set U to be the following set of ordered pairs: ⊆U = def {(A, B) ∈ ℘(U) × ℘(U)  A ⊆ B}
A Relation ‘R’ from a nonempty set A to a nonempty set B is defined as a subset of the Cartesian product set A × B. The subset is derived by defining a relationship between the first element and the second element of the ordered pairs in A × B. In relation R, the set of all first elements is called the domain of the relation R, and the set of all second elements known as images, is called the range of R. For example, the set R = {(1, 2), (– 3, 5), (9, 3)} is a relation; the domain of R = {1, – 3, 9 } and the range of R = {2, 5, 3}. •
A relation can be represented moreover by the set builder form, by the Roster form or by an arrow diagram which is a visual representation of a relation. • If n (A) = x, n (B) = y; then the n (A × B) = xy and the total number of possible relations from the set A to set B = 2xy. In common language, relations are a type of links presented between objects. Examples: ‘neighbor of’ ‘mother of,’ ‘part of,” ‘is an ancestor of,’ ‘is older than,’ ‘is a subset of,’ etc. These are binary relations.
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Properly we will describe relations between elements of sets. We may write Rpq or pRq for “p bears R to q.” And when we define relations as sets of ordered pairs of elements, we will officially write ∈ R. If A and B are any sets and R ⊆ A × B, we denote R a binary relation from A to B or a binary relation between A and B. A relation R ⊆ A × A is known as a relation in or on A. The set dom R = {a ∈ R for some b} is known as the domain of the relation R and the set range R = {b∈ R for some a} is called the range of the relation R.
4.1.3. Functions A relation F between A and B is known as a function from A to B on condition that for every x ∈ A, there occurs an exclusive y ∈ B such that x F y. In this case we write F: A → B. This is frequently expressed by saying that F takes members of A as arguments and returns members of B as values or, otherwise, takes its values in B. Clearly, dom(F) = A For each a ∈ dom(F), the exclusive b such that a F b is known as the value of F at a, written F(a). Consistently, we say F maps a to b, written F: a 7→ b. A relation ‘f’ from a set A to a set B is assumed to be the function if every element of set A has only one illustration in set B. In other words, a function ‘f’ is a relation such that no two pairs in the relation has the same first element. The information f: A →B means that ‘f’ is a function from A to B. A is known as the domain of ‘f’ and B is known as the codomain of ‘f.’ Given an element a ∈ A, there is a unique element ‘b’ in ‘B’ that is related to ‘a.’
The exclusive element ‘y’ to which f relates ‘x’ is denoted by f (x) and is called f of x, or the value of ‘f’ at ‘x,’ or the image of ‘x’ under ‘f.’ The set of all values of f(x) captured together is known as the range of ‘f’ or image of ‘X’ under ‘f.’ Representatively, range of f = {y ∈ Y  y = f (x), for some x in X} A function which has either R or one of its subsets as its range then it is known as a realvalued function. Further, if its domain is also either R or a subset of R, it is known as a real function. A relation F from A to B is a function from A to B, if, and only if it meets both of the following conditions: •
In the domain of F, each element is paired with just one element in the range, i.e., from ∈ F and ∈ F follows that b = c.
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• The domain of F is equal to A, dom F = A. Equivalent definition: A function is defined as a subset R of A × B such that every element of A occurs as the first member of accurately one ordered pair in R. For example, reflect on the sets A = {a, b} and B = {1, 2, 3}. The following relations from A to B are functions from A to B: P = {, } Q = {, } Many terminologies used in discussing about functions is the same as that for relations. We can say that a function having domain A and range a subset of B is a function from A to B, whereas one in A × A is said to be a function in or on A. The notation ‘F: A → B’ is used to denote that ‘F is a function from A to B.’ Elements of the domain of a function are known as the arguments and their correspondents in the range, values. If ∈ F, the familiar notation F(a) = b is used. Commonly used synonyms of function are ‘map’ and ‘mapping.’ A function maps each argument onto an equivalent value. A function F: A → A is also known as an nary operation in A. n
Functions as processes: At times, functions are examined in a different way, as processes, something such as boxes or devices with inputs and outputs. We set the argument in the input and obtain the value of the function in output. In this situation, the set of ordered pairs in the definition is known as the graph of the function. Sometimes partial functions are considered. In this case, second condition in the definition can fall short. In general case, the functions from A to B are said to be into B. If the range of the function equals B, then the function is onto B or an onto mapping or surjection. A function F: A → B is known as onetoone function or injection just in a case when no member of B is assigned to more than one member of A. So, if a≠ b, then F(a)≠ F(b). A function which is onetoone and onto both is further known as a onetoone correspondence or bijection. It is easy to observe that if a function
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‘F’ is a onetoone correspondence, then the relation F–1 is a function and onetoone correspondence. Few specific types of functions: •
•
•
•
•
•
•
Identity function: The function f: R → R which is defined by y = f (x) = x for each x ∈ R is known as the identity function. Domain of f = R Range of f = R. Constant function: The function f: R → R which is defined by y = f (x) = C, x ∈ R, where C is a constant ∈ R, is a constant function. Domain of f = R Range of f = {C} Polynomial function: A real valued function f: R → R which is defined by y = f (x) = a0 + a1 x + . + an x n, where n ∈ N, and a0, a1, a2 an ∈ R, for each x ∈ R, is known as the Polynomial functions. Rational function: These are defined as the real functions of the type f (x)/g (x), where f (x) and g (x) are polynomial functions of x defined in a domain, where g(x) ≠ 0. For example, f: R – {– 2} → R defined by f (x) = 1 2 x x + + , ∀ x ∈ R – {– 2} is a rational function. The Modulus function: The real function f: R → R which is defined by f (x) = x = , 0, 0 x x x x ≥ − 0; 0, if x = 0; –1, if x 2 if and only if there is a nontrivial connection among the producers in B. To show this, assume y0, y1., y n = y0 is a cycle of F(A, B). Then, there are c∈ A, 1 ≤ z ≤ n, such that yz–1czεz = y z where ε z∈ {–1, 1}. Hence, y n = yn–1anεn = yn–2 an–1εn–1cnεn = . = x0 a1ε1c2ε2. cnεn, that is, the identity 1 = c1ε1c2ε2. c n ε n. If this were a trivial relation, then there would be an integer z, 1 ≤ z ≤ n, such that c i = cz + 1 and ε z = εz + 1. However, this implies that yz–1 = yz + 1, a contradiction. Similarly, if c1ε1c2ε2. c n ε n = 1 is a nontrivial 5. L. Babai, Some applications of graph contractions, J. Graph Theory, Vol. 1 (1977) 125–130. 6. L. Babai, Some applications of graph contractions, J. Graph Theory, Vol. 1 (1977) 125–130.
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relation, then y0, y1., yn, where yz = yz–1azεz, 1 ≤ z ≤ n, and y0 = y n, is a closed trial in F(A, B), which must contain a cycle. Suppose now that A is a free group, B a minimal set of producers of A, and B a subgroup of A. Since there is no nontrivial relation on the components of B, F (A, B) does not contain a cycle. Also, from the above outcome, F (A, B) is contractible onto F(X, W) for some set W of generators of X. Because any contraction of a cyclefree graph is again cyclefree, F(X, W) must be cyclefree, and, thus, there is no nontrivial relation on the elements of W. Hence, X must be a free group, freely produced by W.
6.5.4. The SNP Assembly Problem There are many conditions in computational biochemistry where we admire to simplify conflicts between series in a sample by removing some of the series. For sure, accurately what constitutes an argument must be exactly defined in the biochemical context. Graph of conflict is defined where the vertices symbolize the series in the sample, and there is an edge between two vertices if and only if there is a conflict between the matching series. The purpose is to remove the fewest likely series that will remove all conflicts. Recollecting given simple graph F, a vertex cover U is a subset of the vertices such that each edge has at least one end in D. Thus, the purpose is to find a minimum vertex cover in the conflict graph F (in general, this is called a NPcomplete problem7. A particular example of the SNP assembly problem is looked upon as given in 8 and method of problem simplification is shown using the vertex cover algorithm9. A Single Nucleotide Polymorphism (SNP, called as “snip”)10 is a single base alteration in DNA. It is known that SNPs are the most usual source of genetic polymorphism in the human genome (about 90% of all human DNA polymorphisms) (Figure 6.13). 7. R.M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, Plenum Press, 1972. 8. G. Lancia, V. Afna, S. Istrail, L. Lippert, and R. Schwartz, SNPs Problems, Complexity, and Algorithms, ESA 2002, LNCS 2161, pp. 182–193, 2001. SpringerVerlag 2001. 9. Ashay Dharwadker, The Vertex Cover Algorithm, 2006. 10. G. Lancia, V. Afna, S. Istrail, L. Lippert, and R. Schwartz, SNPs Problems, Complexity, and Algorithms, ESA 2002, LNCS 2161, pp. 182–193, 2001. SpringerVerlag 2001.
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Figure 6.13: The DNA double helix and SNP assembly problem. Source: http://www.dharwadker.org/pirzada/applications/figure_1.gif
The SNP Assembly Problem 11 is explained as follows. A SNP assembly is a triple (A, B, C) where A = {a1., an} is a set of n SNPs, B = {b1., b m} is a set of m fragments and C is a relation C: A×B → {0, D, E} indicating whether a SNP ai ∈A does not occur on a fragment b j ∈B (marked by 0) or if taking place, the nonzero value of ai (D or E). Two SNPs a i and a j are termed to be in conflict when there exist two segments b k and b l such that exactly three of R(a i, b k), R(a i, b l), R(a j, b k), R(a j, bl) have the same nonzero value, and accurately one has the differing nonzero value. The issue is to eliminate the fewest possible SNPs that will remove all conflicts. Note, for example, that a1 and a5 are in conflict because R (a1, b2) = E, R (a1, b5) = E, R (a5, b2) = E, R (a5, b5) = D. Again, a4 and a6 are in conflict because R (a4, b1) = D, R (a4, b3) = D, R (a6, b1) = E, R (a6, b3) = D. Similarly, all conflicting SNPs pair are determined easily from the table. 11. G. Lancia, V. Afna, S. Istrail, L. Lippert, and R. Schwartz, SNPs Problems, Complexity, and Algorithms, ESA 2002, LNCS 2161, pp. 182–193, 2001. SpringerVerlag 2001
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The vertex cover algorithm12 is now used to find minimal vertex covers in the conflict graph F. The input is the number of vertices 6, followed by the adjacency matrix of F shown in Figure 6.14. The entry in column j and row i of the adjacency matrix is 1 if the vertices aj and have an edge in the conflict graph and 0 otherwise.
Figure 6.14: The input for the vertex cover algorithm. Source: http://www.dharwadker.org/pirzada/applications/
6.5.5. Computer Network Security A computer scientists team led by Eric Filiol 13 at the Cryptology and Virology Lab, ESAT, and the French Navy, ESCANSIC, have just used the vertex cover algorithm 14to reproduce the proliferation of stealth worms on big computer networks and build optimal strategies for shielding the network against such virus attacks in realtime (Figure 6.15).
12. Ashay Dharwadker, The Vertex Cover Algorithm, 2006. 13 Eric Filiol, Edouard Franc, Alessandro Gubbioli, Benoit Moquet, and Guillaume Roblot, Combinatorial Optimisation of Worm Propagation on an Unknown Network, Proc. World Acad. Science, Engineering, and Technology, Vol 23, August 2007. 14 Ashay Dharwadker, The Vertex Cover Algorithm, 2006.
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Figure 6.15: The set {2, 4, 5} is a minimum vertex cover in this computer network. Source: http://www.dharwadker.org/pirzada/applications/figure_5.gif
The replication was carried out on a big internetlike virtual network and showed that the combinatorial routing topology may have a large impact on the worm proliferation and hence some servers play a more significant and necessary role than others. The realtime potential to recognize them is necessary to greatly restrict worm propagation. The aim is to find a minimum vertex cover in the graph whose vertices are the routing servers and whose edges are the (probably dynamic) links between routing servers. This is the best solution for worm proliferation and the best solution for forming the network defense strategy. Figure 6.15 shows a corresponding minimum vertex cover {2, 4, and 5} and a simple computer network
6.5.6. The Timetabling Problem In a college there are m professor’s y1, y2, …, y m and n subjects x1, x2, …, x n to be taught. Given that professor y i is needed (and able) to teach subject xj for d ij periods (d = [ d ij ] is called the teaching requirement matrix), the college management desires to make a timetable using the minimum possible number of periods. This is called the timetabling problem and can be simplified using the following method. Build a bipartite multigraph G
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with vertices y1, y2, …, ym, x1, x2, …, xn such that vertices yi and xj are connected by dij edges. It is assumed that in any one period every professor can teach at most one subject and that one subject can be taught by at most one professor. First, consider a single period. The single period timetable corresponds to matching in the graph and, on the other hand, each matching symbolizes probable professors assignments related to subjects taught during this period. Thus, the answer to the timetabling problem consists of dividing the G edges into the minimum number of matchings. Equally, the edges of G must be covered with the minimum number of colors. The other method of simplifying this problem is the vertex coloring algorithm. Recollect that the line graph L (F) of F has as vertices the edges of F and two vertices in L (F) are linked by an edge only if the matching edges in F have a common vertex. The line graph L (F) is a simple graph, and a proper vertex coloring of L (F) results in a good edge F coloring using the same color numbers. Therefore, to simplify the timetabling issue, it suffices to find a minimum proper vertex coloring of L (F) utilizing15.
6.5.7. Map Coloring and GSM Mobile Phone Networks Given a map drawn on the sphere surface or the plane, according to the popular four color theorem, it is always possible to color the regions of the map correctly such that no two adjoining regions are allotted the similar color, using maximum four different colors. For any map, we can build its dual graph by following this method. Put a vertex inside every region of the map and link two separate vertices by an edge if and only if their individual regions share a whole segment of their territories in common. Then, a proper coloring of the vertex of the dual graph produces a proper region coloring of the real map (Figure 6.16).
15. Ashay Dharwadker, The Vertex Coloring Algorithm, 2006.
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Figure 6.16: Dual graph of the map of India. Source: http://www.dharwadker.org/pirzada/applications/figure_10.gif
The vertex coloring algorithm is used to properly find a map coloring of India with four colors, see above Figure 6.16. The Groupe Spécial Mobile (GSM) was innovated in 1982 to standardize mobile telephone system. The first GSM network was introduced in 1991 by Radiolinja in Finland with joint technical infrastructure maintenance from Ericsson. At present, the most known standard for mobile phones in the world is GSM, used by more than 2 billion people in more than 212 countries. GSM is a cellular network with its whole geographical range bifurcated into hexagonal cells. Every cell has a communication tower which links with mobile phones within the cell. All mobile phones link to the GSM network by looking for cells in the immediate surroundings. GSM networks work in only four distinct frequency ranges. The reason why only four distinct frequencies suffice is obvious: the cellular regions map can be colored properly by using only four distinct colors! So, the algorithm of
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vertex coloring may be used for allotting at most four distinct frequencies for any GSM mobile phone network, see Figure 6.17.
Figure 6.17: The cells of a GSM mobile phone network. Source: http://www.dharwadker.org/pirzada/applications/figure_11.gif
6.5.8. Knight’s Tours In 840 A.D., alAdli16, a famous shatranj Baghdad (chess) player of Baghdad invented the first reentrant knight’s tour, a progression of moves that takes the knight to every square on an 8×8 chessboard just once, returning to the real square. Many different reentrant knight’s tours were discovered subsequently, but Euler was the first mathematician to do an organized analysis in 1766, not only for the 8×8 chessboard, but for reentrant knight’s tours on the usual n×n chessboard. Given an n×n chessboard, explain a knight’s graph with a vertex matching to each square of the chessboard and an edge connecting vertex i with vertex j if and only if there is a legal move of a knight from the square matching to vertex i to the square matching to vertex j. Thus, a reentrant knight’s tour on the chessboard matches to a Hamiltonian circuit in the knight’s graph. The Hamiltonian circuit algorithm1718 has been used to find reentrant knights tours on chessboards of various dimensions (Figure 6.18).
16. H. J. R. Murray, A History of Chess, Oxford University Press, 1913. 17. Ashay Dharwadker, A New Algorithm for finding Hamiltonian Circuits, 2004. 18. R.M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, Plenum Press, 1972.
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Figure 6.18: A reentrant knight’s tour on the 8×8 chessboard. Source: http://www.dharwadker.org/pirzada/applications/figure_12.gif
6.6. A GRAPH—THEORETIC DATA MODEL FOR GENOME MAPPING DATABASES Graphs are a nice way of presenting genomic data. Graphs have been used to explain gene regulation, mapping relationships, phylogenetic trees, and metabolic pathways. Graphs are a good medium to present mapping information and can be used to explain the concepts involved in laboratory map development and in assisting to carve out the uncertainties and inconsistencies which are contained in published maps. Graph databases are a fine way of querying and storing genome map data. Graphs can explain mapping information and can link that information to the explanation of other genomic data. Rather than translating a map description into objects or relations, a map can be explained directly as a graph. This graph explanation can be stored directly in a database and queries of the database can recover the mapping data. One benefit of storing maps as graphs is that it makes it simple for the user to follow the database structure as the structure is similar to the known map structure.
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6.6.1. Data Models Graphtheoretic data models formalize the manipulation and representation of graph data structures for database systems by defining a movable collection of graphbased type operators and constructors which develop and reach graph data structures. All graphtheoretic data models have the mathematical definition of a graph, i.e., a collection of edges and nodes as their foundation. The data models are normally used to back an application, such as systems19, or extensible databases. A graph database is explained at the theoretical level by a graphtheoretic data model that formalizes the data structures representation stored in a database as a graph. Different from the relational data model which has one embodiment, there are various extensions to the graph definition which form the basis for various graphtheoretic data models. Majority graphtheoretic data models normally add labels to the edges or nodes or both. Often the main graph definition is changed by allowing nodes to sum up graphs and permitting edges to relate to other edges. Linked data models include G + / Graph Log, GOOD, and Hyper log. Graphtheoretic Data model widens the graph definition by including features like: permitting higherorder relations, labeling edges with types and enclosed other graphs as an embodied vertex. The added characteristics make the graphtheoretic data model more helpful than other graphtheoretic data models for structuring genomic data. Operators are described in a graphtheoretic data model which control the graph. Most graphtheoretic data models have operators which delete and add edges and nodes in the graph. Every data model normally adds other operators, like an abstraction mechanism to cluster nodes. A graphtheoretic data model is developed which is energetic toward representing genomic data and consists of an operator for the database querying which is explained below. The database programming language is also developed called WEB which applies some of the operators and can be utilized to develop and query a database graph. WEB is a pronounced programming language like SQL, but depends on a graph logic instead of relational operators. WEB can be directly used, rooted in a different programming language, or accessed via the graphical user interface. A more thorough explanation of operators and graph representation is given which 19 Levene, Marc, and Alexandra Poulovassilis. The hypernode model and its associated query language. In Proc Fifth Jerusalem Conference on Information Technology, pp. 520–530, 1990.
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manipulate them in sections 6.6.3 and 6.6.4.
6.6.2. User View Graphs are the outer view of the database. Each database view is equivalent to the requirements of different tasks or users. No user desires to look at entire graphs for whole genomic data, and no user is interested to sort through a lot of irrelevant information to find the few interesting facts. A scientific database must model various data views. Views communicate to a different task, different users, or different scientific theories. Graphs ease the external views definition as any binary relations collections can be chosen in graph formulation. The significant graphs correspond to concepts in the domain. The external view explains all communications between tire database and the user. Graph database external viewpoints to the graphs which a user can utilize in creating, Manipulating, and querying data. Graphs have been utilized to point schemas at this level using EntityRelation diagrams [S]. Graphs are also used to specify the queries. Querying in a database is performed using a query language which executes some of the conceptual level operators. Query languages are also utilized to develop user views. In a relational database, this is seldom SQL. SQL is a query language that executes some of the relational operators and which the terminal user can either use straightly or implant in a programming language to get the database. Application developers bring SQL to use to define user or external views which permits access to the database part which is important to the application. WEB the Graph logic programming language is used for data entry and querying.
6.6.3. Representation A relation is mathematically defined as a set of ntuples. In graph theory, graphs are defined as a set of edges and vertices G = (V, E). The vertices in a graph mean genomic objects and the edges symbolize binary relationships (roles) between. The basic theoretical definition of a graph has been extended to capture intricacies which take place in genome data. The relationship type between two genomic objects is also crucial; therefore, edge types collections have been added to the graph definition. The edge types point the relation between two vertices or the role that one of them has with respect to the other. For instance, valid relations between
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a YAC and cosmid would include “generated from,” “hybridizes to,” or “shares STS,” or a plasmid could be in relationship to location or sequence objects. The types of the edge are similar to the relations in a relational database constrained to two attributes. Another addition is an abstraction mechanism for graphs which defines enclosed graphs as vertices. These enclosed vertices can be linked to other vertices in a graphlike manner. This method is helpful for symbolizing collected experimental results or reagent libraries. The definition of a graph as a set of edges and vertices is added in our dam model with types of relation which label the edges and an abstraction mechanism which defines other graphs as new vertices in the present graph. Therefore, a graph is a set of: •
Simple vertices which are the nodes of the graph and model the simple concepts and naryl relations of the domain. • Edges which link two vertices. There can be numerous edges between two vertices (with the different or same relation types). • Types of relation which label edges. Therefore, an edge is a relation between one relation type and two vertices. • Graphs which are enclosed into vertices. Each enclosed graph also defines simple edges, vertices, types of relation, and other graphs enclosed into vertices. Another feature added in the graphs is relations between relations (higherorder relations). For example, specifying that one ordering in a radiation hybrid map or genetic linkage is more likely to take place than an alternative ordering. This addition actually eases the database design and as well as increases the graph database value for modeling genomic data. To execute this, Relation type and edges are allowed to be treated as vertices. This results in the following design: •
There are four types of vertices: Packaged graphs, simple vertices, edges, and relation types. • Edges link two vertices and are denoted with a relation type. • Graphs take place within the context of enclosed graphs only. The database contains one enclosed graph which is the basis of the enclosed graph hierarchy. Graph data structure design consists of a labeled, directed, probably cyclic graph which upholds hi erraticallyordered graphs. It also upholds all inverse relationship, positive relationships, and uses of each relationship in an accessefficient data structure.
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The data structure for querying by the partial condition of any graph, and the graph querying algorithm operates by returning all most general graphs in the database which are more specific than the query. Future representation characteristics include calculated arcs whose purpose is calculated for each query and which rely on the current database state.
6.6.4. Operations Operators are required for defining data in a database, upgrading a database, and querying a database. Operators like “project,” “select,” and “join” are defined for a relational database and explain ways via which one can receive and mold relations. Operators must also be defined for receiving and manipulating graphs. As updating operators can be defined in terms of creating and recollecting graphs, operators are described which develop graphs in a database and define algorithms which query graphs in a graph database. There are four important operators on graphs in the graphtheoretic data model. These operators develop graphs in the database, a query against the database or an enclosed graph within it, and remove graphs which are no longer required. Additional operations like inference rules may be added in the future. The operators are: • Feed data using a graph. • Query a graph against the database. • Query a graph against an enclosed graph. • Remove a subgraph from the database. Querying takes place either against an enclosed graph or the whole database. The query operator against the database uses the enclosed graph query operator to query each graph, then mixes the result.
6.7. CASE STUDY: APPLYING GRAPH THEORY TO INTERACTION DESIGN 6.7.1. Introduction A fundamental idea in HCI is that users build mental models of the devices they interact with. Often one can do useful work with quite vague notions of mental and device model, but lowlevel device features have highlevel cognitive effects. For rigorous HCI work, and particularly with safety
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critical devices and tasks, then, it is essential to have a very clear notion of what the device model is. Unfortunately much work in design, specification, and verification of interactive systems uses abstract or incomplete models of devices. What is needed is an approach that can represent full, concrete devices and which has value for the analysis of interaction. If we restrict ourselves to devices that are implemented by computer programs, then the programs (in their given languages) are the final arbiters of the device models. Unfortunately, typical programs do not lend themselves to defining clear device models. Programs (and their specifications) are for instructing computers, not for defining user interface behavior, which in fact happens as a sideeffect of running them. Hardly any code in a typical program has anything explicitly to do with the behavior of the user interface, and typically the code for the user interface is widely distributed throughout the program: there is no single place where interaction is defined. Graphs are a mathematical concept that lends themselves to analysis and interpretation by the program. A large class of interactive system can be built concisely from graphs—and it is a trivial theorem that any digital computer system 2 is isomorphic to a graph and a simple state variable. Significantly, as this paper shows, graphs lend themselves very well to a wide variety of analysis highly relevant to HCI concerns. For example, sequences of user actions are paths in a graph. A standard graphtheoretic concept is the shortest path between two vertices, which defines the most efficient way a user can achieve a particular change of state. If there is no such path, then a user cannot achieve the state change. – The transition matrix M of a graph gives the number of ways a user can cause a state transition by doing exactly one action. The matrix Mn is the P number of ways of achieving any state transition with exactly n actions; and ki = 1 Mi is the number of ways of achieving any transition with 1, 2, 3,… k actions. The higher the number of ways of achieving a state transition, the easier the state is for the user to reach. A safe (a secure interactive device) would typically have only 0 and 1 entries in PMi, whereas a permissive device would have comparatively large entries. In short, graphs very readily simultaneously define interactive systems and usability properties. Graph theory connects formal specification, runnable programs (or prototypes) and HCI. This paper backs up this claim with a wideranging analysis of a working simulation of a real, nontrivial interactive device.
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6.7.2. GraphBased Approaches Although the use of transition systems to specify interactive systems was proposed as early as 1960, they did not catch on as a ‘pure’ formalism because of their apparent limitations for user interface management systems (UIMS)—leading to a line of research that was overtaken by modern rapid application development (RAD) environments. However, the drive behind both UIMS and RAD environments was programmability and flexibility rather than rigor. In rigorous HCI, one needs a programming framework that is both analytic and close to the user interface, if not identical with it: graphs achieve this goal. Graph theory was proposed for use in HCI as a means of analysis; other work includes using graph theory for providing interactive intelligent help and using flow graph concepts to analyze user manuals as structured programs. Graph theory is a substantial area of mathematics, and many interesting theorems and properties are known for graphs that can readily be programmed on a computer. A graph is readily represented by drawing vertices as dots, and arcs as arrows joining dots. Vertex and arc labels are written as words adjacent to the vertices and arcs. If vertices are drawn as circles or other shapes, their labels can be written inside the shapes. Small graphs are easy to draw by hand and larger graphs can be drawn automatically using appropriate tools. To avoid clutter labels are sometimes omitted. Reflexive arcs (also called trivial arcs) that point back to the same vertex are also often omitted for clarity.
6.7.3. Graphs and Interactive Systems We use labeled directed multigraphs in this paper, but what is a graph and how does it relate to an interactive device? A labeled directed multigraph is a set of objects called vertices V, a collection of arcs A ⊆ V × V which are ordered pairs of vertices, and two total functions `V: V → LV and `A: A → LA that map vertices, respectively arcs, to sets of labels, which name the vertices and arcs. The graphtheoretic terms are vertices and arcs, but the device or programming terminology usually refers to vertices as states and arcs as transitions; the user terminology refers to arcs as actions. Formally there is no difference. However, for most devices, the user cannot uniquely identify the state of the device. Instead, the user can observe (hear or feel) indicators. We model this as a mapping O from vertices to the power set of available indicators I, O: V → PPI. That is, in a given state, O(s) is the set of indicators that are ‘shown’ to the user. An interactive device can be represented
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straightforwardly as a directed graph assuming: user actions are mapped into arcs, states are mapped into effects the user can observe (for instance with sounds or indicator lights), and the device must track the current state using a variable. When the user performs an action, the current state A is changed to the next state B where there is a directed arc from A to B labeled with that action. Arcs may point back to the same state, and the transition then does not change the state; if the next state is A, we say that the action is guarded in A as no nontrivial transition occurs. Graph models may be nondeterministic—either because of the underlying system or because of constraints on the modeling process—in which case one of several possible next states will be arrived at. Although useful, nondeterminism complicates many out of our graph metrics, and is beyond the scope of the current paper. Graph models can be extended with other concrete representational details to relate them to actual interactive systems. For example, an image can act as a device’s skin. Changes to the skin during use can be captured by indicator skins— changes to the skin which correspond to the activation of individual indicators. Although an important practical consideration, skins make little impact on our approach. To be formal, devices are considered finite state automata represented by a 10tuple hV, LV, `V, A, LA, `A, O, I, s0, S, IS, iSi, with (in addition to the components already introduced above) s0 the initial state (the state a device is in before it is first used), S the skin (which for our purposes is a color image), and IS a bijection from vertices to indicator skins iS. This level of formality may look pedantic, but there is an important point: precisely this information is sufficient to build a functioning interactive simulation (and even a user manual) and to analyze its usability and other properties in depth. The fruitfulness of this approach is explored throughout this paper. In what follows, we will use the terms state and vertex interchangeably, but stylistically we use the state for userrelated issues and vertex for graphtheoretic four issues. Similarly, we will use action, press, etc., for user actions, but arc for the corresponding graph concept. Typographically, we shall write State and Action.
6.7.4. Example A syringe is used to give patients injections of drugs. A syringe pump is an automatic device that uses a motor to drive the syringe, and gives a patient an injection usually over a period of hours or even days. The pump is set up by a nurse or anesthetist to deliver drugs for various conditions: for example,
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so that it can be used on demand by a patient for pain management. Some pumps have detailed models of drug uptake in the patient (the patient weight having been entered), and may be used for anesthesia. An ambulatory pump is one that a patient can wear or carry around, and is typically used for pain management by delivering calibrated dosages of the drug on demand— within parameters set up by the nurse, particularly so that the patient cannot overdose. This paper uses as a running example a simulation of the main features of the Graseby ambulatory syringe pump type 9500. The simulation of the Graseby pump has been implemented as a Java program, constructed explicitly from a graph model (of 54 vertices and 157 nontrivial arcs)—it is an example of a realisticscale, safety critical interactive system, and thanks to its graphbased definition, with a formal specification that corresponds directly to its interaction behavior. For reasons of space, we only use this one example system; in general, a designer would have a collection of systems and compare properties for variations of the basic design. Clearly, a very important practical use of graph theory is to compare designs, particularly a design and iterative variations of it. For reasons of space, we make no design comparisons here. The remainder of the paper discusses some of the user issues that can be investigated using graph theoretic properties—some of the standard, others of special interest to HCI, and some of the potential opening up new research areas within HCI.
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REFERENCES 1. 2.
3. 4. 5.
Graph Theory Origin and Seven Bridges of Konigsberg. Retrieved from http://www2.gsu.edu/~matgtc/origin%20of%20graph%20theory.pdf Graves, M., Bergemaq, E., & Lawrence, C., (1995). A Graph—Theoretic Data Model for Genome Mapping Databases. Retrieved from https:// www.computer.org/csdl/proceedings/hicss/1995/6921/00/69210032. pdf Pirzada, S., & Dharwadker, A., (2007). Applications of Graph Theory. Retrieved from http://www.dharwadker.org/pirzada/applications/ Rodrigue, D., & Ducruet, D. Graph Theory: Definition and Properties. Retrieved from https://transportgeography.org/?page_id = 5976 Thimbleby, H., & Gow, J. Applying Graph Theory to Interaction Design. Retrieved from http://web4.cs.ucl.ac.uk/uclic/people/j.gow/ papers/eis07.pdf
7 Mathematical Induction and Recursion
CONTENTS 7.1. Introduction .................................................................................... 182 7.2. The Principle of Mathematical Induction......................................... 182 7.3. Proof By Induction: Introduction ..................................................... 184 7.4. Induction And Recursion ................................................................ 195 7.5. Strong Induction ............................................................................. 199 7.6. Case Study: The Flipping Glasses Puzzle ......................................... 203 References ............................................................................................. 209
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Mathematical Induction and Recursion starts with a basic introduction to the topic and then discusses the Principle of Mathematical Induction in detail. After that, ‘Proof by induction’ has been explained and summation part is elaborated for easy understanding with the help of examples. Further, induction and recursion has been explained, and the correlation between both has been discussed in detail. Strong induction, an enhanced version of simple induction has been put in the chapter to broaden the perspective of the readers and provide them with exhaustive information on induction. At last, a case study “The Flipping Glasses Puzzle” has been provided to explain the practical application of Mathematical Induction in the real world.
7.1. INTRODUCTION Above given chapters show the ways to prove the statements which are true for all the things of some: allnatural numbers, all real numbers, all chessboards, etc. Up to now, there are three techniques at the disposal such as proof by contradiction, direct proof and proof by contrapositive. Assuming that we confine ourselves to establishing the facts regarding the natural numbers. There are several good properties in the natural numbers. Some of these properties are like, there no two adjacent natural numbers which have any values between them, each natural number is even or odd, etc. This makes it likely to establish things regarding the natural numbers utilizing methods that do not imply to other structures such as the real numbers, pairs of natural numbers, etc. This particular part explores evidence by induction, a commanding proof method that can be utilized to prove several consequences about natural numbers and discrete structures. We can make use of induction in order to establish some properties regarding natural numbers, to motivate about the accuracy of algorithms, to prove consequences about games, and (later on) to reason about formal models of computation.
7.2. THE PRINCIPLE OF MATHEMATICAL INDUCTION Following given is the definition of the principle of mathematical induction: The principle of mathematical induction:
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“Let P (n) be a property that applies to natural numbers. If the following are true: P (0) is true for any k ∈ ℕ, P (k) → P (k + 1) Then for any n ∈ ℕ, P (k) is true.”
Further given is the brief explanation of the abovegiven definition? Let us suppose that an individual has some property P (n), maybe P (n) is “n is either even or odd,” or P (n) is “the sum of the first n odd numbers is n2.” There are two things which are known about this P (n). Initially, the individual knows that P (0) is true, it means that the property is true when applied to zero. Secondly, we are aware of the fact that if ever we discover that P (k) is true, we will additionally discover that P (k + 1) is true. Hence the next question here is that what would P (n) mean about? Since we are aware of the fact that P (0) is true, we also know that P (1) must be true. Since P (1) must be true, additionally we are aware that P (2) should also be true together with this. P (2) gives us P(3), and subsequently P(3) gives P(4), etc. While actually, it looks like that we must be able to establish that P(n) is true for arbitrary n by making use of the fact that P(0) is true and then displaying P(0), P(1), P(2), …, etc. are all true. According to the principle of mathematical induction certainly, we will be able to accomplish this. In case, we discover any property that initiates with true (p (0) holds and remains to be true when initiated (P(k) → P (k + 1)), now we can summarize that certainly P(n) will be true for all natural numbers n. The method of mathematical induction is comparatively different from rest other techniques which we have studied before. It provides us with a way to highlight that some property is true for every natural numbers n and not straightforwardly displaying that we could incrementally accumulate the consequent one piece at one time. We will be able to find all kinds of example of induction in the real world. Prior to starting working on it through from proofs by induction, let us see if we will be able to build up an intuition for working of induction. Conserving a simple instance like climbing up a flight of stairs. The question here is how will you get on top? Well, the answer lies in the explanation given further. We are aware of the fact that you will only climb up zero steps; subsequently, you can just stand at the base of the stairs and will not reach anywhere. Furthermore, we are aware that in case you’re able to climb zero steps, you must additionally be able to climb one step by climbing zero steps and then taking one step up. We also know that you can
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climb two steps, since you can get up to the first step and then take one step to the second step. In case, you are able to get to the second step, and then you must also be able to reach the third one by taking just one additional step. Following the same procedure again and again, we can display that you can climb till the top of the staircase. We will be able deliberate regarding this inductively as given in the following. Let P (n) be “you can climb to the top of n stairs.” We know that P (0) is true, since you can at all times climb to the top of zero stairs by just not moving. Also, in case you are able to climb to the top of k steps, you can climb to the top of k + 1 steps by merely initiating one more step. In other words, we can say that, for any k ∈ ℕ, P (k) implies P (k + 1). By making use of this particular principle of mathematical induction, you must be able to summarize that you are able to climb a staircase of any height.
7.3. PROOF BY INDUCTION: INTRODUCTION The way to think about the mathematical induction is to favor the statement that we are attempting to establish as not one proposition, but a complete order of propositions, one for each n. The strategy utilized in mathematical induction is to establish the initial given statement in the sequence, and then move on to prove that if any specific statement is true, then all the after ones are also true. This makes us able to determine that all the statements are true. These two can further be explained in a more formal language. The first step: Prove the proposition is true for n = 1. (Or, if the assertion is that the proposition is true for n ≥ a, prove it for n = a.) Inductive step: Prove that if the proposition is true for n = k, then it must also be true for n = k + 1. This particular step is a difficult step, and it may assist if we are dividing it into further several steps. •
•
Stage 1: Write down assertion for the proposition for the case n = k. This is the thing what an individual will assume. It is frequently called the inductive hypothesis. Stage 2: Write down assertion for the proposition for the case n = k + 1. This is the thing an individual need to prove. It should be kept in mind clearly.
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•
Stage 3: After the assumptions are used in stage 1, the statement should be proved in stage 2. The entire explanation regarding the proof is not possible to provide as it varies problem to problem which is dependent upon the content relevant with the mathematical context. You have to use your ingenuity, common sense and knowledge of mathematics here. The question to ask is “how to get started from stage 1 and in what way the person is supposed to reach towards stage 2”? After conducting the inductive step, it can be concluded that the proposition is true for all n ≥ 1 (or for all n ≥ a, if we started at n = a.) For the explanation of abovementioned stages, mathematical induction works best in its own way and techniques. So, in that case, the first thing required to do is to prove the proposition for n = 1. • According to the inductive step, • Since it is true for n = 1, it is also true for n = 2. • Again, by the inductive step, • Since it is true for n = 2, it is also true for n = 3. • And since it is true for n = 3, it is also true for n = 4, and so on. Because we have proved the inductive step, this process will never come to an end. Let say, and there is a number N for which the abovementioned statement is false. So, in that case, if N – 1 is taken into consideration, and the following situation should be taken care of: • The statement is true for n = N − 1, but false for n = N. This contradicts the inductive step, so it cannot possibly happen. Hence the statement must be true for all positive integers n. In the case of a computer programming system, the arguments are comparable, and this comparison could be drawn through a looping process. Under this process, the computation is supposed to be carried out, wherein an indexing variable is advanced by one, and the computation is repeated. These two processes are similar in one way or the other. In the case of a computer program, the first step is to setting an initial value of your variables (this is analogous towards the initial step). Then it is required to set up the loop, calling on the previous values of the variables to calculate new values (this is analogous to our inductive step).
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There is one other thing necessary in a computer program: the programmer is required to set up a “stop” condition; otherwise the program will run forever. That has no analogy in the process, and else the theoretical machine will run forever! That is why the programmer can be certain that the result is true for all positive integers.
7.3.1. Summations For conducting a mathematical induction, simplification of summations is one of the most common applications used for the induction process. For computer programmers and its programming, summations are very frequently used when the programmer is asked to analyze the growth rates of few specific algorithms, in combinatorics when determining how many objects there are of certain sizes, etc. For example, in the case of the selection sort algorithm, an algorithm for sorting a list of values into ascending order. Following observation is required to get started with the process of the algorithm: If the smallest element is eliminated from the list, it would be visible that in the sorted ordering of the list it would appear at the front. Consequently, the programmer would have the flexibility to move that element towards the front of the list, then the remaining would get sorted. After this, the remaining smallest elements of the would be required to be moved towards the secondsmallest position, the smallest of what remains after that to the thirdsmallest position, etc. for example, suppose the given list of numbers is asked to be sorted: 41032 Zero would be removed from the third position, and it will be relocated in the front of the list: 04132 Now, the remaining digits will be sorted, and for that, the smallest element from the list will be traced out of what’s left (the 1), it will be then removed from the list, and would be placed after the 0: 01432 The same process would be repeated to find the smallest and relocating it at the required place, and the result would be: 01243
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After that the smallest value of what remains (3) to get would be moved: 01234 And finally, the last element (4) will be relocated to get the overall sorted sequence: 01234 Whether the abovementioned criterion is efficient or not? To answer this question, the first step would be to quantify the amount of work done to get the list of elements sorted in the right order. Once the quantification process is done, it would become easy to analyze what that quantity is to determine just how efficient the overall algorithm is. Intuitively, the selection sort algorithm works as follows: • While there are still elements left to be sorted; • Scan across all of them to find the smallest of what remains; • Append that to the output. It is quite clear that attaching the smallest remaining value to the output is a time taking process. This is so because, scanning over each element to determine which element is smallest, might take quite a bit of time and could be stressful at the same time. For example, let say, 1 million integers are required to be sorted. By using selection sort, the process would start by scanning over the entire list of one million elements to determine the smallest element from the entire list. Once the determination part is completed, it would become a little easy to scan over the 999,999 remaining elements to determine which of them is the smallest. After that, the remaining 999,998 elements would be sorted to determine which of them the smallest, etc. is. This seems like it’s going to take a while, but just how much time is it going to take? In order to sort a list of n values using selection sort, we need to scan n elements on the first round, then n – 1 on the second, n – 2 on the third, etc. This means that the total number of elements that are supposed to be scanned in the given format: n + (n – 1) + (n – 2) + … + 3 + 2 + 1 What is this value equal to? Well, that depends on the choice of n. In the case of the n values, the following steps would be taken into consideration:
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• When n = 1, the sum is equal to 1. • When n = 2, the sum is 1 + 2 = 3. • When n = 3, the sum is 1 + 2 + 3 = 6. • When n = 4, the sum is 1 + 2 + 3 + 4 = 10. • When n = 5, the sum is 1 + 2 + 3 + 4 + 5 = 15. The main question here is, that is there any scope for any sort of trend or not? Going with the mathematical phenomenon, the answer is a big “yes.” The trend here can be generally described as choosing a right kind of techniques which would help in understanding that in case of a sequence like, {1, 4, 6, 8, 12, ….} what would be the sum. The technique would result in the form of a pattern to understand the way through which the sum would be calculated. For example: Below in Figure 7.1, one of the most frequently used technique is provided which illustrates that an individual could spot something interesting about the problem by drawing a picture (a pictorial representation). Let say, each block represents a number correspondent to a quantity of blocks. Here, 1 signifies one block, 2 signifies 2 blocks, 3 as three blocks, so on and so forth. If all the blocks are placed together next to each other then it could be visualized as shown in Figure 7.1:
Figure 7.1: Visualization of blocks and numbers. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
The above illustration of blocks has resulted in the form of a triangle. In the case of an acute triangle, to calculate the sum the following formula could be used:
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1 bh 2
This formula will help in computing the total area of the triangle. If it is assumed that: n + (n – 1) + … + 2 + 1, Then, in that case, the base of the triangle would be n. According to the formula for the area of a triangle, it would be then expected that the number of blocks could be calculated through a formula given below: number of blocks =
1 2 n 2
But this would not provide with the correct answer. This is so because, the first few values would be 0, 0.5, 2, 4.5, 8, 12.5, … whereas the first few values of the sum of the first n positive natural numbers would be: 0, 1, 3, 6, 10, 15, … But this reasoning is not always suitable. The reason behind this would be that if the real triangle of width n and height n on the top of a boxy triangle gets superimposed, then the result would be (Figure 7.2):
Figure 7.2: Boxy triangle for visualization. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
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From the above Figure 7.2, it is visible that the boxy triangle extends past the bounds of the real triangle by a very slight difference. This illustrates the
1 2 main reason behind having 2 n a value greater than the positive natural number n after summing them up. Although this doesn’t exactly work out correctly, this geometric line of reasoning is actually quite interesting. To 1 calculate the exact area of a triangle which is equal to bh, the belowgiven 2 trivia could be used (Figure 7.3):
Figure 7.3: (a) A triangle: the first step for calculating the area, (b) vertical line drawn as the second step. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
First draw a triangle as given in the above Figure 7.3, then sketch a vertical line downwards through the apex of the triangle as shown in part b of Figure 7.3. After examining the two boxes of the triangle in Figure 7.3 (b), it is clear that half of the area of the box is filled. This illustrates that if the total area of the box is taken into consideration and is the box is cut into two different halves, the area of the triangle could be calculated. In that case area of the triangle could be calculated by the formula: ½bh. Another kind of reasoning that could be used to calculate the sum would be through the given method or technique (Figure 7.4):
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Figure 7.4: From box triangle to a larger box. Source: https://www.people.vcu.edu/~rhammack/BookOfProof/Induction.pdf
The box triangle provided above in Figure 7.4 could be fit into a larger box. In the above Figure 7.4, it is visible that half of the boxes in the large rectangle are related to the sum required to be calculated, while the other half do not. In that case, a rough estimation could be drawn for 1 + 2 + … + n as being about n2. But, Figure 7.4 is not accurate and somewhat unclear as it’s not an even split. For example, in the above Figure 7.4, there are 21 boxes from the
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original sum, and 15 boxes that were added afterward. Although the above drawing doesn’t exactly work, it’s very close to what is exactly required for the calculation purpose. There are several techniques that can be used to fix it. From the above figure, a clear observation could be drawn which states that the boxes that have been added afterward form a triangle of width n – 1 and height n – 1. The initial triangle had the width n and height was n. Given this, suppose that the last column of the triangle is pulled off. Then the picture would be completely different from what it was earlier provided after putting the triangle in the box (as shown in Figure 7.5).
Figure 7.5: After pulling off the last column, the shape of the large triangle box. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
According to the Figure 7.5, the rectangle on the left is somewhat another half of the triangle and this illustrates that half of the boxes are from the original sum and exactly half of the boxes are from the completion. This box has width n – 1 and height n, so of the n (n – 1) total boxes, onehalf of them are from the original sum. Another column which got pulled off from the rectangular large box has n boxes in it. This signifies that 1 + 2 + … + n to be equal to:
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n(n + 1) n(n + 1) 2n n(n + 1) + 2n n(n − 1 + 2) n(n + 1) = +n += = = 2 2 2 2 2 2 Indeed, if the first few terms of n (n + 1)/2 are taken care of, then the sequence would result as: 0, 1, 3, 6, 10, 15, 21. This would match the values for 0, 1, 1 + 2, 1 + 2 + 3, etc. Another interesting way of manipulating the figure would be to change the way the way the extra boxes were added outside the box triangle. If instead of creating a square, a rectangle is created then it would look like the way it is shown in Figure 7.6:
Figure 7.6: Rectangle shaped box when added extra boxes to the triangle. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
The above figure shows that half of the squares in the rectangle are replaced by the triangle. Since this rectangle has area n (n + 1), this means that the sum of the first n positive natural numbers should probably be n (n + 1)/2, which is similar to the previous answer. Another way to have a quick view on the geometric intuition is by abandoning the idea of completing the rectangle. This could be done by superimposing the real triangle width and height n on top of the boxy triangle (as shown in Figure 7.7):
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Figure 7.7: Adding boxed with the same width of the triangle and also the height. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
As before, one idea might be to treat the total area of the boxes as the sum of two different areas – the area covered by the triangle, and the area filled by the pieces of the boxes extending above the triangle. If our triangle has width n and height n, then there will be n number of smaller triangles extending beyond the n by n triangle. Each of these triangles has width 1 and height 1, and therefore has area ½. Consequently, the total area taken up by our boxes would be given by the total area of the large triangle, plus n copies of the smaller triangle. This is
n 2 n n 2 + n n(n + 1) += = 2 2 2 2 And this time the same result has been calculated like before. The main point of understanding here is that there are different ways to solve a problem in mathematics by using different techniques and reasoning power. The result would be similar but the approach is always different, each of which casts light on a slightly different angle of the problem. So, the big question is how this has anything to do with induction at all. Well, at this point we have a pretty good idea that the sum of the
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n(n + 1) first n positive natural numbers is going to be 2 . But how would we rigorously establish this? Here, induction can be invaluable. We can prove that the above sum is correct by showing that it’s true when n = 0, and then showing that if the sum is true for some choice of n, it must be true for n + 1 as well. By the principle of mathematical induction, we can then conclude that it must be true for any choice of n.
7.4. INDUCTION AND RECURSION Mathematical induction and recursion are not very distinct from each other. They are closely connected in mathematical terms and techniques. Mathematical induction is that techniques which could be used and applied to provide a proof about any universal statement for sets of positive integers or their associated sequences, this illustration is provided in such a way that it indicates that there is a meaning to a sentence quantitatively. In the case of recursion, the process of defining an object in its own terms or its terminology through a process of identifying how to solve some simple case of that problem, and solve larger instances of the problem by breaking those instances down into smaller instances. This similarity makes it possible to use induction to reason about recursive programs and to prove their correctness. As an example, let’s take into consideration the following recursive C function, which computes n! In the given way: Int factorial (int n) {if (n = θ) return 1; return n* factorial (n–1)} But, it there a way to give the surety that this actually computes n factorial? After understanding the code provided above, somewhere it is obvious that this might work in the right manner. This could be proved through mathematical induction. In this case, it is first required to show that for any natural number n, that the factorial function, as applied to n, indeed produces n*. This in a sense plays out the recursion backward. The recursive function works by calling itself over and over again with smaller inputs until it reaches the base case. The proof here will be taken into account after understanding the relation of the function with and it’s processing from the bottomup and till then it could be concluded that the factorial function works for the given choice of n.
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Another example will involve the processing of mathematical recursion involving a recursive function which could be applied over lists of elements. There are various programming languages, such as LISP and Haskell that use recursion and lists as their primary means of computation, while other languages like JavaScript and Python support this style of programming quite naturally. In the interests of clarity, rather than writing programs out using any concrete programming language, a pseudocode language will be used that should be relatively easy to read. The main pieces of notation that will be required is given below: •
If L is a list of elements, then L[n] refers to the nth element of that list, zeroindexed. For example, if L = (K, I, L, J, H), then L [0] = K, L [1] = I, etc. • If L is a list of elements, then L[m:] refers to the sub list of L, starting at position m. For example, with L defined as above, L [1:] = (I, H, K, J) and L [2:] = (H, K, J) • If L is a list of elements, then L refers to the number of elements in L. Now, let’s suppose that there is a list of real numbers and it is required to calculate their sum. Going through a recursive method, the problem could be resolved in the following manner: •
The sum of a list with no numbers in it is the empty sum, which is 0. • The sum of a list of (n + 1) numbers is the sum of the first number, plus the sum of the remaining n numbers. Written out in our pseudocode language, we might write this function as follows: Function sum (list L) {if L = θ: return θ else: return L[θ] + sum (L [1:]).} To see how this works, let’s trace the execution of the function on the list (4, 2, 1): Sum (4, 2, 1) = 4 + sum (2, 1) = 4 + (2 + sum(1)) = 4 + (2 + (1 + sum())) = 4 + (2 + (1 + 0))
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= 4 + (2 + 1) =4+3 =7 There is a reason behind the successful processing of the recursive method. The reason is that, every time when the sum function is applied, the size of the list shrinks by one element. But, the list cannot be shrined for forever, so eventually the base case will be targeted. The processing could be proven by going through the previous recursive function. It is important to prove the function and its accuracy, because the argument to the function was itself a natural number. Now, the argument to the function is a list of values. Fortunately, though, this does not end up causing any problems. Although the actual list itself is not a natural number, the length of that list is a natural number. Therefore, it is important to prove the correctness of the algorithm by showing that it works correctly for lists of any length. This trick by using induction on the size or shape of some object helps in enabling the user to use recursion to prove results that don’t directly apply to the natural numbers. After this explanation, this can be said that there are ways to prove the correctness of functions that execute over lists. For further explanation, we can discuss one function that operates over sums, which has been mentioned below, function product (list L) { if L = 0: return 1. else: return L[0] × product(L[1:]). } To test the execution efficiency of the function mentioned above, let’s take an example of a small list, the one with the values (2, 3, 5, 7) product (2, 3, 5, 7) = 2 × product (3, 5, 7) = 2 × (3 × product (5, 7)) = 2 × (3 × (5 × product (7)))
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= 2 × (3 × (5 × (7 × product ()))) = 2 × (3 × (5 × (7 × 1))) = 2 × (3 × (5 × 7)) = 2 × (3 × 35) = 2 × 105 = 210 The proof of this function is proved correct and it is similar to proving our sum function correct, as the properties of these two functions have comparatively similar recursive structure. The reason that the proofs discussed above are very similar has been discussed in further sections, which will be discussed in later sections, and is not a mere coincidence. Let’s consider one more example to strengthen our understanding. Let’s consider that we have we have a list of real numbers and want to return the maximum value contained in the list. For example, max (1, 2, 3) = 3 and max (π, e) = π. To maintain the parity, let’s define that the maximum value of an empty list is ∞. The following function can be written to compute the maximum value of a list as follows: Function listMax (list L) { if L = 0: return ∞. else: return max (L [0], listMax (L [1:])). } It is interesting that the functions we have just discussed look almost similar and if we trace out the execution of these programs, it will end up giving the similar results to the one we have discussed before: listMax (2, 3, 1, 4) = max (2, listMax (3, 1, 4)) = max (2, max (3, listMax (1, 4))) = max (2, max (3, max (1, listMax (4)))) = max (2, max (3, max (1, max (4, listMax ())))) = max (2, max (3, max (1, max (4, ∞))))
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= max (2, max (3, max (1, 4))) = max (2, max (3, 4)) = max (2, 4) =4 To provide the proof of this function is quite easy, as we have already mentioned the same proof earlier in the sections discussed above.
7.5. STRONG INDUCTION The multiple versions of induction have been discussed so far like normal induction or induction starting at k – have broaden our perspectives and provided a variety of useful results. This section discusses an even more exhaustive form of mathematical induction called Strong Induction. Strong induction has multiple applications in the fields of computer science and technology, ranging from understanding of the structure of numbers themselves to the analysis of algorithms. Before discussing the Strong Induction in detail, let’s revise a little about how normal induction works. In a proof by induction, at first, we will discuss that some of the properties hold true for 0, or some other numerical value as discussed before. Then we will conclude that “since it’s true for 0, it’s true for 1,” then conclude “since it’s true for 1, it’s true for 2,” then conclude “since it’s true for 2, it’s true for 3,” etc. It can be worth noticing that at every step, only the most recent results have been used, that have been obtained in order to get to the next result. That is, to prove that the outcomes are true for the cases mentioned above, only the use of the facts that the results is true for two, and not that it is true for one or zero. Similarly, if we take a large number in order to prove the results for example, say 137, the proof will only be dependent on the fact that the result was true for 136. In other words, we are limiting ourselves and are handicapping ourselves. The intuition behind the Strong induction was the need to delimit ourselves and why shall we be depended only on the most recent results that were proven by us. The entire set should be brought into use in order to create the next results. Theorem (strong induction): Let P(n) be a property that applies to natural numbers. If the following are true: P(0) is true.
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For any k, if ∈ ℕ P(0), P(1), …, P(k) are true, then P(k + 1) is true. Then for any n ∈ ℕ, P(n) is true.
When we compare the strong induction with the normal principle of mathematical induction, it has been noted that, we used to assume that P (k) is true, then we will use it show that P (k + 1) is true. And on other hand, we used to assume that all of P (0), P (1),… and P(k) are true, and use this to show that P(k + 1) is true. To give a simple hint of what Strong Induction is, let’s consider one simple example that will tell how to implement this style of proof technique in the reallife application. For example, suppose that one person has a chocolate bar that consists of k + 1 smaller squares of chocolate, all arranged in one line. The candy bar might look like as shown below in Figure 7.8.
Figure 7.8: Pictorial depiction of a candy bar. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
Further, if we want to break this chocolate bar into smaller parts and create n + 1 squares. The question is how many breaks are required to make in the chocolate bar in order to completely break it down into the desired number of pieces? Let’s try to understand the concept with the help of examples. If one has a chocolate candy bar with six pieces, then a total of five breaks are required to accomplish the job – one in between each of the pieces of chocolate. Further, if the chocolate candy bar has 137 pieces, 136 beaks are needed to accomplish the job. In general terms, it seems like one needs to break the chocolate candy bar n times if there are n + 1 squares, as there are n separators. The results cannot be that interesting, and does not seem that impressive, but if we want to prove that this is the optimal solution, a bit more care is
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required in this case. It’s definite that the candy bar can be broke with n breaks, but can it be done by using n breaks? The answer shall be no, and to prove this, the following theorem is required: Theorem: Breaking a linear candy bar with n + 1 pieces down into its individual pieces requires at least n breaks. The main point is, how exactly this theorem can be discussed. In order to prove this theorem, it is required to show that, under any circumstances, if we need to break the candy bar apart, one always need to make at least n breaks. The following can be done by presenting a line of reasoning, that “consider any possible way to break apart the candy bar, then show that no matter how it’s done, it always uses at least n breaks.” So, what is the difference in breaking the candy in this way? If we think over it, in any way, if we break apart the candy bar, it must start with some initial break that will break the candy bar into smaller pieces. From this point, we can further break the candy into smaller parts, and these smaller parts can be further divided into even smaller pieces, for instance, below mentioned is a way in which the candy bar can be divided into six smaller pieces (Figure 7.9).
Figure 7.9: Breakdown of candy bar for strong induction. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
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Now, let’s discuss the key insights drawn from the above discussion: it can be noticed that when we beak the chocolate bar with the first break, we get the two smaller chocolate bars. In order to break the overall chocolate bar down into several individual squares, it is required that we need to break those smaller pieces down into their individual parts. Simultaneously, we can opt for any approach for breaking the chocolate bar as mentioned below: • •
Firstly, make some initial break beak in the chocolate bar. Secondly, break the remaining pieces of chocolate down into their constituent pieces. And then, at some point, this process has to terminate, and indeed it can be taken note of that once when we get down to a chocolate bar of size one, there is no nothing much left to do. By considering these insights, we can use strong induction and prove that the total number of breaks required to break the candy bar is at least n. To do so, we need to prove the base case at first, which states that a chocolate bar with only one piece will require no further breaks, and from this, it can be concluded that no matter how you break the chocolate bar into pieces, the total number of breaks that will be needed to subdivide the remaining pieces, along with the initial break, is always at least n, in case the chocolate bar has n + 1 pieces in it. For now let’s discuss the proof in brief, which is further explained in detail. Proof: Let’s discuss the proof by Strong Induction. Let P (n) be “breaking a linear chocolate bar with n + 1 pieces down into its constituent pieces requires at least n breaks.” The aim is to prove P (n) for all n ∈ ℕ by strong induction on n.
For the base case that we are trying to prove, we need to show that P (0), that any method of breaking a chocolate bar having a single square into its constituent parts having at least zero breaks. This case is true, as if there is
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only one square, it is already split down into smaller parts as it can be, that requires no breaks. For our inductive step, it has been assumed that for some k ∈ ℕ, that for any k’ ∈ ℕ with k’ ≤ k, that P (k’) holds and breaking a candy bar with k’ + 1 pieces into its squares, which takes at least k’ breaks. P (k + 1), that breaking a candy bar with K + 2 pieces that require at least k + 1 breaks. To bring this into account, it shall be noted that anyway, we can break this candy bar and it will consist of an initial break that will split the candy bar into two pieces, followed by subsequent breaks of those smaller candy bars. Assume that we break the candy bar in a way that there is m + 1 small squares left in one smaller piece and (k + 2) – (m + 1) = (k – m) + 1 pieces in the second piece. Here, the condition is that m + 1 should not be greater than k + 1, as if it were, it would have k + 2squares in one smaller piece and 0 in other smaller piece, that means that we actually did not break anything. This means that m + 1 < = k + 1, so that m < = k. Thus by bringing in action our strong inductive hypothesis, it is known that a minimum of m breaks are required to split any piece into m + 1 pieces that are constituent in nature. In a similar way, since m > = 0, we know that k – m ≤ k, so by putting the inductive hypothesis a minimum of at least k – m breaks are required to break the piece of size (k – m) + 1 into its constituent pieces. This shows that for any initial break, the total number of breaks that will be needed is at least (k – m) + m + 1 = m + 1, as required. Thus, it can be seen that P(m + 1) hold true, which n turn completes the induction. In conclusions, it can be said that, before we claim the fact that inductive hypothesis can be used on the smaller pieces, we need to verify that the size of every small piece shall not be greater than k. It is important that the method discussed above shall be used in any proof, when strong induction is being implemented. The inductive hypothesis can only be applied to natural numbers that are less than or equal to k, and if one wants to apply the inductive hypothesis to something of size n,’ you need to first show that k’< = k.
7.6. CASE STUDY: THE FLIPPING GLASSES PUZZLE Consider the following puzzle: you are given five wine glasses, as shown here (Figure 7.10):
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Figure 7.10: Glasses as discussed in puzzle. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
You want to turn all of the wine glasses upsidedown, but in doing so are subject to the restriction that you always flip two wine glasses at a time. For example, you could start off by flipping the first and last glasses, as shown here (Figure 7.11):
Figure 7.11: First and last glasses inverted. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
And could then flip the second and fourth glasses, like this (Figure 7.12):
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Figure 7.12: Second and fourth glasses inverted. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
From here, we might flip the first and third glasses to get this setup (Figure 7.13):
Figure 7.13: First and third glass inverted. Source: https://web.stanford.edu/class/cs103/notes/Mathematical%20Foundations%20of%20Computing.pdf
If you play around with this puzzle, though, you’ll notice that it’s tricky to get all of the wine glasses flipped over. In fact, try as you might, you’ll never be able to turn all of the wine glasses over if you play by these rules. Why is that? Figuring the answer out requires a bit of creativity. Let’s count how many wine glasses are facing up at each step. Initially, we have five wine glasses facing up. After our first step, we flip two of the wine glasses,
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so there are now three wine glasses facing up. At the second step, we have several options: • • •
Flip over two glasses, both of which are facing up; Flip over two glasses, both of which are facing down; or Flip over two glasses, one of which is facing up and one of which is facing down. How many wine glasses will be facing up after this step? In the first case, we decrease the number of wine glasses facing up by two, which takes us down to one glass facing up. In the second case, we increase the number of wine glasses facing up by two, which takes us to five glasses facing up. In the third case, the net change in the number of wine glasses facing up is zero, and we’re left with three glasses facing up. At this point we can make a general observation – at any point in time, each move can only change the number of upfacing wine glasses by + 2, 0, or –2. Since we start off with five wine glasses facing up, this means that the number of wine glasses facing up will always be exactly 1, 3, or 5 – all the odd numbers between 0 and 5, inclusive. To solve the puzzle, we need to get all of the wine glasses facing down, which means we need zero wine glasses facing up. But this means that the puzzle has to be impossible, since at any point in time the number of upwardfacing wine glasses is going to be odd. The question now is how we can formalize this as a proof. Our argument is the following: • •
The number of upwardfacing wine glasses starts off odd. At any point, if the number of upwardfacing wine glasses is odd, then after the next move the number of upwardfacing wine glasses will be odd as well. • The argument here is inherently inductive. We want to prove that the number of glasses starts odd, and that if it starts odd initially, it will stay odd forever. There are many ways to formalize the argument, but one idea would be to prove the following for all natural numbers n: Lemma: For any natural number n, after n moves have been made, the number of upwardfacing glasses is an odd number. The phrasing here says that no matter how many moves we make (say, n of them), the number of upward facing glasses will be odd. Given this lemma, it’s extremely easy to prove that the puzzle is unsolvable. So how
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exactly do we prove the lemma? In a proof by induction, we need to do the following: •
Define some property P (n) that we want to show is true for all natural numbers n. • Show that P (0) is true. • Show that for any natural number k, if P (k) is true, then P (k + 1) is true as well. Let’s walk through each of these steps in detail. First, we’ll need to come up with our property P (n). Here, we can choose something like this: Let P (n) be the statement “After n steps, there are an odd number of upwardfacing glasses.” Notice that our choice of P (n) only asserts that there are an odd number of upwardfacing glasses for some specific n. That is, P (4) just says that after four steps, there are an odd number of upwardfacing glasses, and P (103) just says that after 103 steps, there are an odd number of upwardfacing glasses. This is perfectly normal in an induction proof. Mathematical induction lets us define properties like this one, then show that the property is true for all choices of natural numbers n. In other words, even though we want to prove that the claim is true for all natural numbers n, our property only says that the claim must be true for some specific choice of n. Now that we have our property, we need to prove that P (0) is true. In this case, that means that we have to show that after 0 steps, there are an odd number of upwardfacing glasses. This is true almost automatically – we know that there are five upwardfacing glasses, to begin with, and if we make 0 steps we can’t possibly change anything. Thus there would have to be five upwardfacing glasses at the end of this step, and five is odd. It seems almost silly that we would have to make this argument at all, but it’s crucial in an inductive proof. Remember that induction works by showing P(0), then using P(0) to get P(1), then using P(1) to get P(2), etc. If we don’t show that P (0) is true, then this entire line of reasoning breaks down! Because the entire inductive proof hinges on P (0), P (0) is sometimes called the inductive basis or the base case. When writing inductive proofs, you’ll often find that P (0) is so trivial that it’s almost comical. This is perfectly normal, and is confirmation that your property is not obviously incorrect. Always make sure to prove P (0) in an inductive proof. The last step in induction is to show that for any choice of k ∈ ℕ, that if P (k) is true, P (k + 1) must be true as well. Notice the structure of what we need to show – for any choice of k we must show that P (k) implies P (k + 1).
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As you saw the last chapter, to prove something like this, we’ll choose some arbitrary natural number k, then prove that P(k) → P(k + 1). Since our choice of n is arbitrary, this will let us conclude that P(k) → P(k + 1) for any choice of k. In turn, how do we then show that P(k) → P(k + 1)? This statement is an implication, so as we saw the last chapter, one option is to assume that P(k) is true, then to prove P(k + 1). This step of the proof is called the inductive step, and the assumption we’re making, namely that P(k) is true, is called the inductive hypothesis. If you think about what we’re saying here, it seems like we’re assuming that for any k, P(k) is true. This is not the case! Instead, what we are doing is supposing, hypothetically, that P(k) is true for one specific natural number k. Using this fact, we’ll then go to show that P(k + 1) is true as well. Since the statements P(k) and P(k + 1) are not the same thing, this logic isn’t circular. So we now have the structure of what we want to do. Let’s assume that for some arbitrary natural number k ∈ ℕ, that P(k) is true.
This means that after k steps, the number of upwardfacing glasses is odd. We want to show that P(k + 1) is true, which means that after k + 1 steps, the number of upwardfacing glasses is odd. How would we show this? Well, we’re beginning with the assumption that after k steps there are an odd number of upwardfacing glasses. Let’s call this number 2m + 1. We want to assert something about what happens after k + 1 steps, so let’s think about what that (k + 1)st step is. As mentioned above, there are three cases: •
We flip two upwardfacing glasses down, so there are now 2m + 1–2 = 2(m – 1) + 1 upwardfacing glasses, which is an odd number. • We flip two downwardfacing glasses up, so there are now 2m + 1 + 2 = 2(m + 1) + 1 upwardfacing glasses, which is an odd number. • We flip one upwardfacing glass down and one downwardfacing glass up, which leaves the total at 2m + 1 upwardfacing glasses, which is also odd. So in every case, if after k steps the number of upwardfacing glasses is odd, then after k + 1 steps the number of upwardfacing glasses is odd as well. This statement, combined with our proof of P (0) from before, lets us conclude by mathematical induction that after any number of steps, the number of upwardfacing glasses is odd.
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REFERENCES 1.
2. 3.
4.
5.
6.
Barnes, M., & Gordon, S., (1987). Mathematical Induction. [eBook] Mathematics Learning Centre, University of Sydney. Available at: https://sydney.edu.au/stuserv/documents/maths_learning_centre/ induction.pdf [Accessed 23 August 2018]. Mathematical Induction, (n.d.). [eBook] Available at: http://www. tkiryl.com/teaching/aa/review1.pdf [Accessed 23 August 2018]. Mathematical Induction, (n.d.). [eBook] Available at: https://www. people.vcu.edu/~rhammack/BookOfProof/Induction.pdf [Accessed 23 August 2018]. O’Connor, J., (2009). Mathematical Induction Sequences and Series. [eBook] Available at: http://wwwhistory.mcs.stand.ac.uk/~john/ MISS.pdf [Accessed 23 August 2018]. Proof by Mathematical Induction, (n.d.). [eBook] Available at: http:// web.maths.unsw.edu.au/~jim/proofsch8.pdf [Accessed 23 August 2018]. Schwarz, K., (2015). Mathematical Foundations of Computing. [eBook] Available at: https://web.stanford.edu/class/cs103/notes/Mathematical Foundations of Computing.pdf [Accessed 23 August 2018].
8 Cardinality
CONTENTS 8.1. Introduction .................................................................................... 212 8.2. What Is Cardinality ......................................................................... 212 8.3. Types Of Cardinality ....................................................................... 213 8.4. Types Of Sets .................................................................................. 215 8.5. Subsets ........................................................................................... 222 8.6. Sets With The Same Cardinality....................................................... 223 8.7. Set Theory Symbols ......................................................................... 224 8.8. Boolean Algebra ............................................................................. 226 8.9. Values Of Cardinality ...................................................................... 227 8.10. Elementary Theorems .................................................................... 228 8.11. Advanced Theorems ...................................................................... 229 8.12. Cardinality Of The Continuum ...................................................... 230 8.13. Controversies ................................................................................ 230 8.14. Conclusion ................................................................................... 233 References ............................................................................................. 235
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About the labeled chapter, we shall look into an introduction regarding what mathematics is and how it is used in our daytoday life. We will also be getting in detail overview about what cardinality is, what the various types of cardinality are. The below also gives an insight about the types of sets into detail. Elementary theorems, advanced theorems have also a given us a wide knowledge about what can be derived from the different types of sets. There are also various controversies that various mathematicians put forth in contrast to the already existing theorems.
8.1. INTRODUCTION The term ‘Mathematics’ is derived from the Ancient Greek μάθημα (máthēma) which means, “That which is learnt” or “what one wants to know.” Mathematics in the initial periods was used to designate any subject of instruction or study. Mathematics which is concerned with the investigation of mathematical concepts which originally arisen from space and numbers. In the later centuries and as learning advanced, it was found to be quietly suitable and restricted the scope of the study and its term to a particular field of knowledge. It is normally accepted that mathematics originated with the handson problems of counting and recording numbers. The beginning of the inkling of number is so hidden behind the veil of countless ages that it is tantalizing to speculate on the remaining indications of early humans’ sense of number. Our remote ancestors of some 20,000 years ago—who were quite as clever as we are—must have felt the need to enumerate their livestock, tally objects for barter, or mark the passage of days. However, the evolution of counting, with its spoken number words and written number symbols was gradual and does not allow any determination of precise dates for its stages. Here, we are going to see about one of the greatest inventions in mathematics, the Cardinality, its uses and theorems in detail.
8.2. WHAT IS CARDINALITY Word Origin: 1520s. “The condition of being a Cardinal,” from Cardinal (n.) + ity. Its mathematical sagacity is from 1936 when math became the property of possessing a cardinal number. The term cardinality refers to the
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number of cardinal members in a set. Cardinality can be infinite or finite. For example, the cardinality of the set of people in the European Countries are approximately 4300, 000,000; the cardinality of the set of integers is denumerable infinite.
8.3. TYPES OF CARDINALITY The various types of cardinality, in general, are as follows: • • •
cardinality of database; cardinality of SQL numbers; and cardinality of sets.
8.3.1. Cardinality of Database In database designing, cardinality plays a very important concept. Many different types of cardinality exist in Mathematics, and they need to be used correctly to properly design database. Cardinalities are used to create an E/R diagram that shows the relationship between entities and tables. The first type of cardinality is basic 1:1 (one to one). This is where we might have one car driver for every car. The second type of cardinality is a 1: M (one to many). It means that one car has many mechanics. The third type of cardinality is an M: N (many to many). This means that many mechanics work on many cars. However, with an M: N relationship it is more realistic to turn it into two 1: M relationships. A person can accomplish the desired result by adding an extra table in between mechanic and car named “works on.” These are sometimes referred to as dummy tables. We would say that each mechanic works on a car and a mechanic works on each car. The last type of cardinality is an M: N: M (many to many to many). This is only used in a subtype/supertype situation. This is a hieratical type model, and the primary key must be found in all of the characteristics ranging from the very top to bottom. Here we could start with a supertype of person and then move down to subtypes of racers and mechanics, and then subtype of those could be pro and amateur. All with the race team being included in all of their attributes.
8.3.2. Cardinality of SQL Numbers In a SQL (Structured Query Language), the term cardinality refers to the exclusivity of data values contained in a particular column (attribute) of a database table. The lower the cardinality, the more duplicated elements in a
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column. Thus, a column with the lowest possible cardinality would have the same value for every row. SQL databases use cardinality to help determine the optimal query plan for a given query.
8.3.3. Cardinality of Sets Set: “A set is a welldefined collection of distinct objects. The objects of a set are called its elements.” Element of a Set: An object or idea in a set is called an element (or member) of the set. Notation: The symbol ∈ is used to denote that an element is a member of a set and ∉ is used to denote that an object is not a member of a set. Example: For set A = {1, 2, 3}, 1 ∈A, but 12 ∉A. Example: For S = {x: x is a state bordering Minnesota}, Iowa ∈S, but Alabama ∉S.
Cardinality of sets is the most important and widely used cardinality in mathematics. In the below table, the number of rows (or tuples) is called the cardinality. Practically, tables always have positiveinteger cardinality. The reason for this is simple: tables with no rows, or with a negative number of rows, cannot exist in theory. An example for a table with denumerable infinite cardinality is a multiplication table of nonnegative integers in which entries are implied for all possible values.
0 1 2 3 :
1 1 2 3 :
2 2 4 6 :
3 3 6 9 :
. . . .
The concept of cardinality has been used to demonstrate that some infinite sets are larger than others are. The cardinality of the set of real numbers is always greater than the cardinality of the set of integers, even though both sets are infinite. The cardinality of the set of integers is called alephnull or alephnaught; the cardinality of the set of real numbers is called alephone. One of the greatest mysteries of mathematics is, “What is the cardinality of the set of points on a geometric line?” Commonly, it is presumed to be alephone; the set of points on a line is thought to correspond onetoone
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with the set of real numbers. This is by no means a trivial supposition, and has become known as the Continuum Hypothesis. Cardinality is the general term for the size of a set, whether finite or infinite. The term cardinality refers to the number of basic members in a set. Cardinality can be finite (a nonnegative integer) or infinite. The cardinality of a set is denoted by S. It was the work of Georg Cantor (1845–1918) to establish the field of set theory and to discover that infinite sets can have different sizes. Initially, his work was controversial, but that swiftly became a foundation to modern mathematics. Cardinality provides an overview to the surprising notion of “uncountable sets”: infinite sets with so many elements that it is impossible to make a list x1, x2, x3. of all of the m (even if the list is allowed to be infinitely long).
8.4. TYPES OF SETS • • • • • • • •
Finite set; Infinite set; Null set; Union set; Intersection set; Equal set; Equivalent set; and Universal set.
8.4.1. Finite Sets The cardinality of a finite set is simply the number of elements in the set. For example, the cardinality of {1, 2, 3} is 3, and the cardinality of the empty set is 0. The set A has cardinality n (for nonnegative integer n) if there is a bijection (onetoone correspondence) between the set {1, 2, 3,…, n} and the set A. Validly then we define the bijection f:{1, 2, 3}→{a, b, c} by f(1) = a, f(2) = b, and f(3) = c, thus proving {a, b, c} has cardinality 3., We can symbolize that the cardinality of the set A by A. This looks like “absolute value,” and it measures the size of a set, just as absolute value measures the magnitude of a real number. Another useful notation that does not appear in our book is to let [n] be the set {1, 2, 3,…n} with [0] = ∅. Then we can state that A = n if there is a bijection f:[n]→A.
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The below are the few examples of finite sets: • Let P = {10, 50, 65, 70, 15, 40} Then, P is a finite set and n(P) = 6. • Let Q = {natural numbers less than 25} Then, Q is a finite set and n(P) = 24. • Let R = {whole numbers between 5 and 45} Then, R is a finite set and n(R) = 38. • Let S = {x: x ∈ Z and x^2–81 = 0} Then, S = {–9, 9} is a finite set and n(S) = 2. • •
The set of all persons in Australia is a finite set. The set of all birds in India is a finite set.
8.4.2. Infinite Sets An infinite set A is countably infinite if there is a bijection f: ℙ →A, where ℙ is the set of positive integers. That stands ℙ = {1, 2, 3,…}. A set is countable if it is finite or countably infinite. A synonym for countable is denumerable. Infinite sets that are not countable are uncountable or, less frequently, no denumerable. Below are the few examples of infinite sets: • • • •
Set of all points in a plane is an infinite set. Set of all points in a line segment is an infinite set. Set of all positive integers which is multiple of 3 is an infinite set. W = {0, 1, 2, 3, …….} i.e., set of all whole numbers is an infinite set. • N = {1, 2, 3, ……….} i.e., set of all natural numbers is an infinite set. • Z = {……… –2, –1, 0, 1, 2, ……….} i.e., set of all integers is an infinite set. Thus, from the above discussions, we know how to distinguish between the finite sets and infinite sets with examples.
8.4.3. Null Set The null set, also called the empty set, is the set that does not comprise anything. It is symbolized or { }. There is only one null set because there is reasonably only one way that a set can contain nothing.
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The null set makes it possible to openly define the results of operations on certain sets that would otherwise not be openly definable. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set. For example: {1, 3, 5, 7, 9.} {2, 4, 6, 8, 10.} = The null set provides a basis for construction of a formal theory of numbers. In mathematics, zero is defined as the cardinality of (which is, the number of elements in) the null set. From this initial point, arithmeticians, and mathematicians can shape the set of natural numbers, and from there, the sets of integers and rational numbers. Examples of the null set: • The set of whole numbers less than 0. • Undoubtedly there is no whole number less than 0. Hence, it is an empty set. Questions regarding the null set: The question is: Are all empty sets equal? No matter the circumstances. For example is {x: x is positive integer less than zero} equal to {x: x is an integer between 9 and 10} Answer: Yes, all empty sets are equal. Let us see an instance to understand better. Denote A = {x: x is positive integer less than zero} and B = {x: x is an integer between 9 and 10} So, I claim that A = B. If you do not agree with me, you have to show that A is different from B. To do so, you have to demonstrate me an element in one set that does not belong to another set. Can we do that? Can we point to an offending element if both sets have no elements whatsoever? Indeed, can we say that a “positive integer less than zero” which is not an “integer between 9 and 10”? Of course, we cannot, because there are no positive integers less than zero. Can we say that an “integer between 9 and 10” which is not a “positive integer less than zero”? Of course, we cannot, because there are no integers between 9 and 10. Hence you cannot prove that A is not equal to B. Therefore you have to agree with me that A = B.
8.4.4. Union of Set Union of two given sets is the smallest set, which contains all the elements of both the sets.
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To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated. The symbol for denoting the union of sets is ‘∪.’ For example; Let set A = {2, 4, 5, 6} and set B = {4, 6, 7, 8} Taking every element of both the sets A and B, without repeating any element, we get a new set = {2, 4, 5, 6, 7, 8} The new set contains all the elements of set A and all the elements of set B with no recurrence of elements and is named as the union of set A and B. The symbol used for the union of two sets is ‘∪.’ Thus, representatively, we write union of the two sets A and B is A ∪ B that means A union B (Figure 8.1).
Figure 8.1: A∪B. Source: https://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Union_ of_sets_A_and_B.svg/371pxUnion_of_sets_A_and_B.svg.png
Therefore, A ∪ B = {x: x ∈ A or x ∈ B}
Properties of the operation of union sets: • • • • •
A∪B=B∪A
(Commutative law)
A∪φ=Α
(Law of identity, φ is the identity of Union)
U∩A=A
(Law of U)
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(Associative law)
A∪A=A
(Idempotent Law)
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Notes: A ∪ ϕ = ϕ ∪ A = A, i.e., union of any set with the empty set is always the set itself.
8.4.5. Intersection Sets From two sets A and B, the intersection is the set that contains elements or objects that belong to A and to B at the same time. The intersection operation has several obvious properties: • Commutativity: A∩B = B∩A. • Associativity: (A∩B)∩C = A∩(B∩C). • A∩B = A if, and only if, A⊆ B. We write A ∩ B
We find A ∩ B by looking for all the elements A and B have in common.
For example: Let A = {1 orange, 1 pineapple, 1 banana, 1 apple } and B = { 1 spoon, 1 orange, 1 knife, 1 fork, 1 apple } A ∩ B = {1 orange, 1 apple}
Let A = {1 orange, 1 pineapple, 1 banana, 1 apple } and B = { 1 spoon, 1 orange, 1 knife, 1 fork, 1 apple } A ∩ B = {1 orange, 1 apple} (Figure 8.2)
Figure 8.2: A∩B. Source: https://commons.wikimedia.org/wiki/File:Intersection_of_sets_A_ and_B.svg
Thus, A ∩ B
Properties of the operation of Intersection sets: If A, B, and C are three sets, then
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A∩B=B∩A
(A ∩ B) ∩ C = A ∩ (B ∩ C)
(Commutative law) (Associative law)
f∩A=A
(Law of identity, φis the identity of U) (Idempotent Law)
U∩A=A
(Law of U)
A∪A=A
8.4.6. Equal Set
Two sets A and B can be equal only if each element of set A is also the element of the set B. In addition, if two sets are the subsets of each other, they are said to be equal. This is represented by: A=B A ⊂ B and B ⊂ A ⟺ A = B
If the state seen above is not met, then the sets are said to be unequal. This is represented by A≠B For instance, If P = {1, 3, 9, 5,−7} and Q = {5,−7, 5, 3, 1, 9,−7}, then P = Q. This is for the reason that no matter how many times an element is repeated in the set, it is only counted once. In addition, the order does not matter for the elements in a set. Therefore, to rephrase in terms of cardinal number, we can say that If A = B, then n(A) = n(B) and for any x ∈ A, x ∈ B too.
8.4.7. Equivalent Sets
To be an equivalent set, the sets should have the same cardinality. This means that there should be one to one correspondence between the elements of both the sets. Now, one to one correspondence means that for each element in the set A, there exists an element in the set B until the sets get exhausted. Definition 1: If two sets A and B have it cardinality, if there exists an objective function from set A to B. Definition 2: Two sets A and B are said to be equivalent if they have the same cardinality, i.e., n(A) = n(B).
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Examples of Equal and Equivalent sets If P = {1,−7, 200011000, 55} and Q = {1, 2, 3, 4}, then P is equivalent to Q If C = {x: x is an integer} and D = {d: d is a natural number}, then C is equivalent to D Some important points: • • i. ii.
All the null sets are equivalent to each other. If A and B are two sets such that A = B, then A is equivalent to B The above means that the two equal sets will always be equivalent, but the converse of the same may or may not be true. Not all infinite sets are equivalent to each other. For e.g., the set of all real numbers and the set of integers.
8.4.8. Universal Set A universal set does not have to be the set of everything that is known or thought to exist – such as the planets, extraterrestrial life, and the galaxies – even though that would be one of the examples of a universal set. A universal set is all the elements, or members, of any group under consideration. For instance, all the stars of the Milky Way galaxy could be a universal set if we are conferring all the stars of the Milky Way galaxy. This type of universal set might be appropriate for astronomers, but it is still a large set of objects to consider. A characteristic universal set in mathematics is the set of natural numbers as shown below: N = {1, 2, 3, 4.}. Boldface capital letters are occasionally used to identify certain number sets, such as N for natural numbers. We usually use braces to enclose a set. The ellipsis mark (.) tells us that the set of natural numbers goes on without end, so this universal set is also an infinite set. However, universal sets can also be finite sets. Let us look at the case of a universal set that is finite. The set of all the presidents of Russia is an example of a universal set that is finite. This set may increase every three years, but at any given time, it is a finite universal set if we are discussing all the men who have been president of Australia. Symbols in Universal Sets: Sets are generally named with a capital letter. Therefore, the universal set is typically named with the capital letter U. This will be the notation used in this lesson.
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Sometimes, alternate symbolization might be used for a universal set, such as the example of the set of natural numbers. The set of natural numbers is not necessarily a universal set. Whether a set is a universal set is based on the structure of a problem or on the situation under examination. But the point that states here is that alternate notation can be used to designate a universal set as long as it is practical and clear to the observer.
8.5. SUBSETS If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B, and we write it as A ⊆ B or B ⊇ A The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in.’ Some facts about Subsets:
• Every set is a subset of itself, i.e., A ⊂ A, B ⊂ B. • Empty set is a subset of every set. • Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in.’ • A ⊆ B means A is a subset of B or A is contained in B. • B ⊆ A means B contains A. For example:
• Let A = {2, 4, 6} • B = {6, 4, 8, 2} Here A is a subset of B. Since, all the elements of set A are contained in set B. But B is not the subset of A. Subsequently, all the elements of set B are not contained in set A. Notes: If ACB and BCA, then A = B, i.e., they are equal sets. Every set is a subset of itself.
8.5.1. Super Set Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A. Symbol ⊇ is used to denote ‘is a superset of’ For example:
A = {a, e, i, o, u} B = {a, b, c., z}
Here A ⊆ B, i.e., A is a subset of B but B ⊇ A, i.e., B is a superset of A
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8.5.2. Proper Subset If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A, i.e., A ≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B. For example:
A = {1, 2, 3, 4} Here n(A) = 4
B = {1, 2, 3, 4, 5} Here n(B) = 5 We observe that, all the elements of A are present in B, but the element ‘5’ of B is not present in A. So, we say that A is a proper subset of B. Symbolically, we write it as A ⊂ B Notes:
No set is a proper subset of itself. Null set or ∅ is a proper subset of every set.
8.5.3. Power Set The group of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set. For example: If A = {p, q} then all the subsets of A will be P(A) = {∅, {p}, {q}, {p, q}} Number of elements of P(A) = n[P(A)] = 4 = 22 Overall, n[P(A)] = 2m where m is the number of elements in set A.
8.6. SETS WITH THE SAME CARDINALITY The usual symbol for the cardinality of ℙ is 0 א, which is read “alephnaught.” Aleph is a letter of the Hebrew alphabet. This is the smallest infinite cardinal. The next are א, 1 א2, etc. Thus the commencement of the list of cardinal numbers looks like 0, 1, 2, 3,…, 2 א, 1 א, 0 …א. That is, it is an infinite list followed by further infinite lists. There is a standard to denote the cardinality of ℝ by c (for continuum). A person can derive that c is greater than א0, but it is unknown whether c = 1 א. This proposition (that c = 1 )אis called the continuum hypothesis. It
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has been shown in the 20th century to be independent of the axioms of set theory, which means that it does not contradict them nor can it be proved from them. This gives it a status the same as that of the parallel postulate in geometry: we can accept it or deny it and get a consistent axiomatic system either way. Sets with the Same Cardinality have the same cardinal numbers if there exists a bijection between them. The balance of the section is mostly obscure. The two results worth noting are the following: If A and B set and there exist injections f:A→B and g:B→A, then A and B have the same cardinality. This is occasionally easier than finding a bijection between A and B. For instance, you can map ℤ to 2 ℤ (the even integers) by mapping n to 2n, and you can map 2 ℤ to ℤ by mapping n to (1/2)n. Both maps are injections, so  ℤ = 2 ℤ .
8.6.1. Result
There is no bijection between a finite set and a proper subset of that set. This is not really what the theorem talks about, but it is probably the part of the theorem that is truly useful. Note that this theorem fails for infinite sets. For instance, 2 ℤ is a proper subset of ℤ, but there is an obvious bijection between them, as described above.
8.7. SET THEORY SYMBOLS List of set symbols of set theory (Table 8.1). Table 8.1: Table of Set Theory Symbols Symbol
Symbol Name
Meaning/ definition
Example
{}
Set
A collection of elements
A = {3, 7, 9, 14}, B = {9, 14, 28}

Such that
So that
A∩B
Intersection
Objects that belong to set A and set B
A∪B
Union
Objects that belong to set A or set B
A⊆B
Subset
A is a subset of B. set A is included in set B.
A = {x  x∈
, x