Proof, Logic, and Conjecture: The Mathematician's Toolbox [1 ed.] 0716730502, 9780716730507

Table of contents :
Frontmatter
Table of Contents
Preface
Unit 1 Logic and Proofs
Chapter 1 Introduction
1.1 Knowledge and Proof
Proofs in Science
Proofs in Law
1.2 Proofs in Mathematics
Discovery and Conjecture in Mathematics
Organization of the Text
Chapter 2 Propositional Logic
2.1 The Basics of Propositional Logic
2.2 Conditionals and Biconditionals
Biconditionals
2.3 Propositional Consequence; Introduction to Proofs
Chapter 3 Predicate Logic
3.1 The Language and Grammar of Mathematics
3.2 Quantifiers
3.3 Working with Quantifiers
Negations of Statements with Quantifiers
Some Abbreviations for Restricted Quantifiers
3.4 The Equality Relation; Uniqueness
Uniqueness
Chapter 4 Mathematical Proofs
4.1 Different Types of Proofs
Formal Proofs
A General-Purpose Axiom System for Mathematics
Informal Proofs
Good Proofs
4.2 The Use of Propositional Logic in Proofs
4.3 The Use of Quantifiers in Proofs
Counterexamples
Some Theorems Involving Quantifiers
4.4 The Use of Equations in Proofs
Doing the Same Thing to Both Sides of an Equation
Reversibility
4.5 Mathematical Induction
Axioms for the Natural Numbers
The Meaning of Mathematical Induction
The Structure of Proofs by Mathematical Induction
Mathematical Discovery Revisited
4.6 Hints for Finding Proofs
Gaining Insight into a Proof
Unit 2 Sets,Relations, and Functions
Chapter 5 Sets
5.1 Naive Set Theory and Russell’s Paradox
Naive Set Theory
The Paradoxes of Set Theory
5.2 Basic Set Operations
Subsets, Proper and Otherwise
The Sum Rule for Counting
5.3 More Advanced Set Operations
Indexed Families of Sets
Unions and Intersections of Collections of Sets
Chapter 6 Relations
6.1 Ordered Pairs, Cartesian Products, and Relations
Relations
Inverse Relations
6.2 Equivalence Relations
*6.3 Ordering Relations
Preorderings
Irreflexive Orderings
Chapter 7 Functions
7.1 Functions and Function Notation
Function Notation
Why Codomains?
Other Ways of Defining Functions
7.2 One-to-One and "Onto" Functions; Inverse Functions and Compositions
Compositions
Inverse Functions
Restricting the Domain of Functions
7.3 Proofs Involving Functions
Guidelines for Proving Things about Functions
Induced Set Operations
Inverse Images
7.4 Sequences and Inductive Definitions
Definitions by Induction
Justification of Inductive Definitions
7.5 Cardinality
Finite and Infinite Sets
Countable and Uncountable Sets
7.6 Counting and Combinatorics
Permutations
Combinations
7.7 The Axiom of Choice and the Continuum Hypothesis
The Axiom of Choice
The Continuum Hypothesis
Unit 3 Number Systems
Chapter 8 The Integers and the Rational Numbers
8.1 The Ring Z and the Field Q
The Field of Rational Numbers
8.2 Introduction to Number Theory
Complete Induction and the Fundamental Theorem of Arithmetic
Some Well-Known Theorems of Number Theory
Some Famous Conjectures of Number Theory
*8.3 More Examples of Rings and Fields
Modular Arithmetic
“Well-Definedness” of Operations on Quotient Structures
Finite Fields
*8.4 Isomorphisms
Chapter 9 The Real Number System
9.1 The Completeness Axiom
Completeness of the Real Number System
9.2 Limits of Sequences and Sums of Series
Infinite Series and Decimals
9.3 Limits of Functions and Continuity
Continuity
*9.4 Topology of the Real Line
Connected Sets
*9.5 The Construction of the Real Numbers
The Construction of the Integers and the Rationals
Chapter 10 The Complex Number System
10.1 Complex Numbers
10.2 Additional Algebraic Properties of C
Appendices
Appendix 1 A General-Purpose Axiom System for Mathematics
Rules of Inference!
Axioms
Footnotes
Appendix 2 Elementary Results About Fields and Ordered Fields
The Field Axioms
Proofs from the Field Axioms
Proofs Using the Ordered Field Axioms
Appendix 3 Some of the More Useful Tautologies
Solutions and Hints to Selected Exercises Index Errata
References
List of Symbols and Notation
Index

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