Discrete Mathematics [Lam Rfc Cr ed.] 1423224884, 9781423224884

Table of contents :
Discrete Math
Logic of Statements
Arguments
Quantified Statements
Methods of Proof
Proving Universal Statements
Sequences
Mathematical Induction
Sets
Counting
Functions & Set Relations
Graphs & Trees

Citation preview

BarCharts, Inc.®

WORLD’S #1 ACADEMIC OUTLINE

Logic of Statements Statements & Logical Connectives

A statement is a sentence that is either true or false, but not both.

Logic, Graphs & Trees, Sets, Probability EX: Construct a truth table for the compound statement ~p ∨ (q ∧ r).

EX: The moon orbits the Earth is a true statement. EX: One plus one equals three is a false statement.

A compound statement is a statement that is made up of one or more simple statements using logical connectives. For statements p and q, the table summarizes the logical connectives that can be used to create compound statements.

Logical Connectives Expression ~p p ∧ q p ∨ q

How it is read “not p” “p and q” “p or q”

Type of connective negation conjunction disjunction

Compound statements may contain multiple logical connectives. For compound statements that contain a negation and another logical connective, the negation (~) is performed first. EX: ~p ∨ q is equivalent to (~p) ∨ q. EX: ~p ∧~q is equivalent to (~p) ∧ (~q).

You can express compound statements symbolically using ~, ∧, and ∨.

EX: Let p represent the statement “It is cloudy.” Let q represent the statement “It is raining.” Represent each compound statement below symbolically. It is not raining. ~q It is cloudy and it is raining. p∧q It is cloudy but it is not raining. p ∧ ~q or p ∧ (~q) It is not cloudy and it is not raining. ~p ∧ ~q or (~p) ∧ (~q)

Truth Tables

A statement is either true or false and has a well-defined truth value. Compound statements also have well-defined truth values that are determined by the truth values of the simple statements that comprise them. Truth tables can be used to show the truth value of a compound statement based on the truth values of its components. EX: Negation of a statement For a statement p, the negation of p is written ~p and means “not p.” The truth value of ~p is the opposite of the truth value of p. If p is true, then ~p is false. If p is false, then ~p is true.

T F

F T

q

T T F F

T F F F

q

T T F F

T F T F

q∧r

~p ∨ (q  ∧  r )

T T T T F F F F

T T F F T T F F

T F T F T F T F

F F F F T T T T

T F F F T F F F

T F F F T T T T

Two statements are logically equivalent if and only if they have identical truth values. The symbol ≡ is used to denote logical equivalence. Steps to Determine Logical Equivalence For compound statements P and Q: 1. Construct a truth table for P. 2. Construct a truth table for Q using the same statement variables as step 1. 3. Compare the truth values of the tables. If the truth values in each row are the same, the statements are logically equivalent. If the tables have a different truth value in any row, the statements are not logically equivalent. De Morgan’s laws are used to determine the negation of a conjunction and the negation of a disjunction. The negation of a conjunction is logically equivalent to a disjunction in which each component is negated. ~(p ∧ q) ≡ ~p ∨ ~q EX: Find the negation of the statement Justine is a sophomore and has blonde hair. The negation of this statement is Justine is not a sophomore or does not have blonde hair.

The negation of a disjunction is logically equivalent to a conjunction in which each component is negated.

~(p ∨ q) ≡ ~p ∧ ~q EX: Find the negation of the statement Luis was sick on Monday or he missed the school bus. The negation of this statement is Luis was not sick on Monday and he did not miss the school bus.

p

T T F F

q

T F T F

~p F F T T

~q

(p ∨ q)

F T F T

T T T F

~ (p ∨ q) F F F T

~ p ∧ ~q F F F T

Conditional Statements

A conditional statement is an if-then statement of the form “if p, then q,” where p is the hypothesis and q is the conclusion. This is symbolized by p → q or read as “p implies q.”

EX: Disjunction of a statement For statements p and q, the disjunction of p and q is written p ∨ q and means “p or q.” The truth value of a disjunction is true when at least one of p or q is true. If both p and q are false, then the disjunction is false.

p

~p

EX: Construct a truth table to show that ~(p  ∨  q) ≡ ~p  ∧ ~q.

p∧q

T F T F

r

Logical Equivalence & De Morgan’s Laws

EX: Conjunction of a statement For statements p and q, the conjunction of p and q is written p ∧ q and means “p and q.” The truth value of a conjunction is true when, and only when, both p and q are true. If either p or q is false, or if both p and q are false, the conjunction is false.

p

q

As with other compound statements, a truth table can be used to demonstrate the logical equivalence of De Morgan’s laws.

~p

p

p

EX: If 12 is divided by 4, then the quotient is 3. If Nathan makes the soccer team, then he will play fullback. If Christy does not earn at least a B on her final, then she will earn a C for the class. A conditional statement is false when, and only when, the hypothesis p is true and the conclusion q is false. Otherwise, the conditional statement is true.

p∨q T T T F

Truth tables can also be used to determine the truth values of more complicated compound statements.

p

q

p→q

T

T

T

T

F

F

F

T

T

F

F

T

Truth tables can be used to analyze compound conditional statements. 1

Logic of Statements (continued )

A conditional statement is logically equivalent to its contrapositive. This can be demonstrated with a truth table.

EX: Construct a truth table for the conditional statement (p ∨ ~q) → ~q.

p

q

~q

(p ∨ ~q)

(p ∨ ~q) → ~q

T T F F

T F T F

F T F T

T T F T

F T T T

p

T T F F

Negation of a Conditional Statement

The negation of the conditional statement “if p then q” is logically equivalent to “p and not q.” In symbols this is ~(p  → q) ≡ p  ˄ ~q.

q

T F T F

~p F F T T

~q

~q  → ~p

p  → q

F T F T

T F T T

T F T T

The converse of a conditional statement of the form p → q or “If p, then q” is q → p or “If q, then p.” EX: Write the converse of the conditional statement. • Statement: If tomorrow is the 31st of the month, then today is the 30th. Converse: If today is the 30th of the month, then tomorrow is the 31st. The inverse of a conditional statement of the form p → q or “If p, then q” is ~p → ~q or “If not p, then not q.” EX: Write the inverse of the conditional statement. • Statement: If a number is divisible by 4, then it is divisible by 2. Inverse: If a number is not divisible by 4, then it is not divisible by 2. NOTE: A conditional statement and its inverse are NOT logically equivalent. Also, a conditional statement and its converse are NOT logically equivalent.

EX: Write the negation of the conditional statement. • Statement: If Karl is elected to Student Council, then he will be treasurer. Negation: Karl is elected to Student Council, and he is not treasurer.

Contrapositive, Converse & Inverse

The contrapositive of a conditional statement of the form p → q or “If p, then q” is ~q  → ~p or “If not q, then not p.” EX: Write the contrapositive of each conditional statement. • Statement: If it is raining, then there are clouds in the sky. Contrapositive: If there are no clouds in the sky, then it is not raining. • Statement: If a number is an integer, then the number is rational. Contrapositive: If a number is irrational, then the number is not an integer.

Arguments Premises & Conclusions

In mathematics, an argument is a sequence of statements. All of the statements except for the final one are called premises. The final statement is called the conclusion of the argument. The symbol ∴ is typically placed in front of the conclusion of an argument and means “therefore.” EX: The following argument has 2 premises and a conclusion. If a quadrilateral is a square, then the quadrilateral is a rectangle. Quadrilateral ABCD is a square. ∴ Quadrilateral ABCD is a rectangle.

Arguments can also be represented in symbolic form.

EX: In the previous example, let p = a quadrilateral is a square and let q = the quadrilateral is a rectangle. The argument has the following symbolic form. p → q p ∴q

Valid & Invalid Arguments

An argument is valid when regardless of what particular statements are substituted for the premises, if the resulting premises are all true, then the conclusion is also true. If the argument does not have a valid form, it is said to be invalid. To test an argument for validity, follow these steps: 1. Identify the premises and conclusion of the argument and represent them with symbols. 2. Make a truth table with a column for each premise and a column for the conclusion.

3. Find the rows in which all of the premises are true. These are called the critical rows. 4. If the conclusion is true for each critical row, then the argument is valid. If there are any critical rows in which the conclusion is false, then the argument is invalid.

EX: Test the validity of the following argument by constructing a truth table. Craig was on the red team or Craig was on the blue team. Craig was not on the red team. Therefore, Craig was on the blue team. p ∨ q ~p ∴q

p

q

T T F F

T F T F

p  ∨ q T T T F

~p F F T T

q T F T F

There is 1 critical row in which both of the premises are true. Because the conclusion in this row is also true, this is a valid argument.

Modus Ponens

The argument and its symbolic form shown are valid. If you have a current password, then you can log on to the school network. You have a current password. Therefore, you can log on to the school network.

p → q p ∴q

This argument form is called modus ponens which is Latin for the “method of affirming.” A truth table can be used to show that an argument of this form is valid.

p

T T F F

p → q

q

T F T F

T F T T

p

T T F F

q

T F T F

Modus Tollens

The argument and its symbolic form shown are valid. If it is raining, then there are clouds in the sky. There are no clouds in the sky. Therefore, it is not raining. p → q ~q ∴ ~p

This argument form is called modus tollens which is Latin for the “method of denying.” A truth table can be used to show that an argument of this form is valid.

p

T T F F

q

T F T F

p → q T F T T

~q F T F T

~p F F T T

Quantified Statements Predicates, Domains & Truth Sets

A predicate is a sentence that contains a finite number of variables. A predicate becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all possible values that can be substituted for the variable. EX: Let P(x, y) = “x > 10 and x + y = 20.” The predicate is P, the variables are x and y, and the domain of the variables is all real numbers. When x = 12 and y = 8, P(12, 8) = “12 > 10 and 12 + 8 = 20.”

If P(x) is a predicate and has domain D, then the truth set of P(x) is all x∈D such that P(x) is true. The truth set of P(x) is denoted {x ∈ D | P( x )} which is read “the set of all x in D such that P(x).” EX: Let P(x) be “x is a factor of 12” and let D be the set of all positive integers. The truth set of P(x) is {1, 2, 3, 4, 6, 12}.

Universal Statements & Quantifiers

A predicate can be changed into a statement by assigning specific values to each of its variables. Another way to change a predicate into a statement is to add quantifiers. Quantifiers are words that refer to some or all of a set and tell for how many elements of the set the given predicate is true. The universal qualifier is represented by the symbol ᗄ which means “every” or “for all.” For a given predicate P(x) where x has domain D, a universal statement has the form ᗄx∈D, P(x). A universal statement is defined to be true if and only if P(x) is true for every x in D. It is defined to be false if there is at least one element in D for which P(x) is false. The element is called a counterexample to the universal statement. EX: Let D = {2, 3, 4, 5}. Then the universal statement ᗄx∈D, x2 > x + 1 is true because 22 > 2 + 1, 32 > 3 + 1, 42 > 4 + 1, and 52 > 5 + 1.

2

EX: Let Z+ = all positive integers. Then the universal statement ᗄx∈ Z+, x 2 > x + 1 is false because a counterexample can be found: x = 1 is a counter­ example because 1∈ Z +, but 12 1 + 1.

Existential Statements & Quantifiers The existential qualifier is represented by the symbol ∃ which means “there exists.” For a given predicate P(x) where x has domain D, an existential statement has the form such ∃ x∈D that P(x). An existential statement is defined to be true if and only if P(x) is true for at least one x in domain D. It is false if and only if P(x) is false for all values of x in domain D. EX: Let Z = the set of integers. Then the existential statement ∃ x∈Z such that x(x + 1) = 30 is true because 5∈ Z and 5(5 + 1) = 30.

Quantified Statements (continued )

Methods of Proof

Negations of Quantified Statements

The negation of a universal statement is an existential statement, and the negation of an existential statement is a universal statement. For the universal statement of the form ᗄx∈D P(x), the negation is the existential statement ∃ x∈D such that ~P(x). For the existential statement ∃ x∈D such that P(x), the negation is the universal statement ᗄx∈D, ~P(x). EX: Write the negation of each statement. • Statement: All of the pens in a desk drawer are blue. Negation: There is a pen in the desk drawer that is not blue. • Statement: There is a red convertible among the cars in a parking lot. Negation: For all of the cars in a parking lot, none of them are a red convertible. • Statement: ᗄ composite numbers x, x is even. Negation: ∃ x∈ composite numbers such that x is

not even.

• Statement: ∃ a right angle with a measure not equal to 90°. Negation: ᗄ right angles the angle has a measure equal to 90°.

Universal Conditional Statements

The universal conditional statement has the form ᗄx∈D, if P(x) then Q(x). EX: For all polygons, if a polygon is a rectangle, then the polygon is a parallelogram.

The negation of the universal conditional statement of the form ᗄx∈D, if P(x) then Q(x) is the statement ∃ x∈D such that P(x) and ~Q(x). EX: Write the negation of the universal conditional statement. • Statement: ᗄ positive integers x, if x is divisible by 5, then x is divisible by 10. Negation: ∃ x∈ positive integers such that x is divisible by 5 and x is not divisible by 10.

Definitions

An integer is even if and only if it can be expressed as 2k for some integer k.

EX: • 14 is even because it can be expressed as 2(7). • –100 is even because it can be expressed as 2(–50). • If m and n are integers, then 14m2n is even because it can be expressed as 2(7m2n). NOTE: 7m2n is an integer because integers are closed under multiplication.

An integer is odd if and only if it can be expressed as 2k + 1 for some integer k.

EX: • 23 is odd because it can be expressed as 2(11) + 1. • –17 is odd because it can be expressed as 2(–9) + 1. • If m and n are integers, then 4m + 6n + 1 is odd because it can be expressed as 2(2m + 3n) + 1.

All positive integers greater than 1 are either prime or composite. An integer, n, is prime if and only if n > 1 and for all positive integers a and b, if n = a ∙ b, then a = 1 or b = 1. An integer, n, is composite if and only if n = c ∙ d for some positive integers c and d with c ≠ 1 and d ≠ 1.

EX: • 11 is prime because its only factors are 1 and 11. • 9 is composite because it can be written as the product of factors 3 ∙ 3.

A real number is rational if and only if it can be written as Any real number that is not rational is called irrational. EX: • 5 is rational because it can be written as

a b

for some integers a and b where b ≠ 0.

5 . 1

10 • 0.03300330033 . . . is rational because it can be written as 303 . π • 2 is irrational because it cannot be expressed as the ratio of 2 integers.

If c and d are integers with d ≠ 0, then c is divisible by d if and only if c = d ∙ k for some integer k. The notation d | c means that c is divisible by d and is read “d divides c.” The integer c is a multiple of d, and d is a factor or divisor of c. EX: • 91 is divisible by 7 because 91 = 7 ∙ 13. • 6 | 42 because 42 = 6 ∙ 7.

Proving Existential Statements An existential statement of the form ∃ x∈D such that P(x) is true if and only if P(x) is true for at least one element of the domain. A constructive proof of existence can be used to prove an existential statement. In a constructive proof of existence, an element x in D is found and shown to make P(x) true. EX: Prove the following statement: ∃ a prime number, p, between 34 and 40. Proof: Let p = 37. 37 is between 34 and 40, and 37 is prime.

Proving Universal Statements A universal statement of the form ᗄx∈D, if P(x) then Q(x) is true if and only if the conditional statement is true for each value of x in the domain. Several methods exist for proving universal statements.

Proof: Suppose a, b, and c are arbitrary integers where a | b and b | c. Then by the definition of divisibility, b = a ∙ m and c = b ∙ n for some integers m and n. Substitute the expression for b into the expression for c and simplify: c=b∙n c = (a ∙ m) ∙ n c = a ∙ (m ∙ n) Let k = m ∙ n. Since m and n are integers, k is an integer, and c = a ∙ k. Therefore, by the definition of divisibility, a | c.

Method of Exhaustion

If the domain D is finite, the method of exhaustion can be used to prove a universal statement. With the method of exhaustion, each element of the domain is shown to result in a true conditional statement. EX: Prove the following statement: ᗄ integers n, if 1 ≤ n ≤ 5, then n2 – n + 11 is a prime number. Proof: 12 – 1 + 11 = 11 is a prime number 22 – 2 + 11 = 13 is a prime number 32 – 3 + 11 = 17 is a prime number 42 – 4 + 11 = 23 is a prime number 52 – 5 + 11 = 31 is a prime number

Proof by Contradiction

A statement is either true or false. It cannot be both. If the assumption that a particular statement is false logically leads to a contradiction, then the assumption must be false, and the original statement must be true. This is the basis for proof by contradiction. Steps for a Proof by Contradiction 1. Assume that the statement to be proven is false. 2. Show that this assumption logically leads to a contradiction. 3. Conclude that the original statement must therefore be true.

Direct Proof

Many universal statements are made about domains with an infinite number of elements which precludes using the method of exhaustion. The method of direct proof is often used to prove universal statements. To apply this method, follow the steps outlined below. 1. Express the statement to be proved in the form: ᗄx∈D, if P(x) then Q(x). 2. Suppose x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true. 3. Use definitions, logic, and established results to show that Q(x) is also true.

EX: Prove: The difference of any rational number and any irrational number is irrational. Suppose there is a rational number m and an irrational number n such that m – n is rational. a

c

By the definition of rational numbers, m = b and m – n = for integers a, b, c, d and d with b ≠ 0 and d ≠ 0. Substitute for m: c

a

c

m–n= d b –n= d Solve for n:

EX: Prove: If n is an odd integer, then (–1)n = –1. ᗄ integers n, if n is odd, then (–1)n = –1. Proof: Suppose n is an arbitrary odd integer. Then by the definition of odd, n = 2k + 1 for some integer k. Substitute: (–1)n = (–1)2k + 1. Simplify: (–1)2k + 1 = (–1)2k(–1)1 = (–1)2k(–1) = ((–1)2)k(–1) = (1)k(–1) = (1)(–1) = –1. Therefore, (–1)n = –1 for odd integers n. EX: Prove: For integers a, b, and c, if a divides b and b divides c, then a divides c. ᗄ integers a, b, and c, if a | b and b | c, then a | c.

c ad-bc b – d = n bd = n a

Because integers are closed under multiplication and subtraction, b ≠ 0, and d ≠ 0, ad – bc is an integer and bd is a non-zero integer. So, n is the ratio of two integers and is rational which contradicts the assumption that n is irrational. Therefore, the original statement must be true. It can be concluded that the difference of any rational number and any irrational number is irrational. 3

Sequences

Mathematical Induction

Definitions

A sequence is a set of objects (such as numbers) written in a specific order. Each object is called a term of the sequence. EX: The sequence 10, 8, 6, 4, 2 is made up of 5 terms: 10, 8, 6, 4, and 2.

In general, a numerical sequence can be defined as ak, ak + 1, ak + 2, … , am, where each individual number ai is a term of the sequence. Sequences can be made up of a finite number of terms or an infinite number of terms.

EX: The sequence 1, 4, 9, 16, 25, 36, 49 is a finite sequence with 7 terms. EX: The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … is an infinite sequence because it contains an infinite number of terms.

An explicit formula for a numerical sequence is a rule that gives a particular term ak based on the value of k. EX: The explicit formula ak = k + 2k defines a sequence. Find the first 5 terms of the sequence. a1 = 1 + 21 = 1 + 2 = 3 a2 = 2 + 22 = 2 + 4 = 6 a3 = 3 + 23 = 3 + 8 = 11 a4 = 4 + 24 = 4 + 16 = 20 a5 = 5 + 25 = 5 + 32 = 37

Summation Notation

It is often desirable to find the sum of the terms of a sequence, a1 + a2 + a3 + a4 + a5 + . . . , however it is not convenient (or even possible sometimes) to list all of the terms of the sequence. Summation notation is a shorthand method of expressing the sum of a finite or infinite sequence. The capital Greek letter sigma, ∑, is used to represent such sums as follows. n

∑a k =1

k

= a1 + a2 + a3 + … + an – 1 + an 5

EX: Compute the sum

∑ (2k + 3) . k =1

5

∑ (2k + 3) = [2(1) + 3] + [2(2) + 3] + [2(3) + 3] + [2(4) + 3] + [2(5) + 3] = 5 + 7 + 9 + 11 + 13 = 45 6

k =1

EX: Compute the sum 6

∑ (−1)

k

∑ (−1)

k

.

k =1

= (–1)1 + (–1)2 + (–1)3 + (–1)4 + (–1)5 + (–1)6 = –1 + 1 + (–1) + 1 + (–1) + 1 = 0

k =1

Product Notation

Similar to summation notation, product notation is a shorthand method of expressing the product of the terms of a sequence. The capital Greek letter pi, ∏, is used to represent a product of terms as follows. n

Π a = a1 ∙ a2 ∙ a3 ∙ … ∙ an – 1 ∙ an

k =1 k

4

EX: Compute the product Π (k + 1) . 4

k =1

Deductive Reasoning & Inductive Reasoning Deductive reasoning, or deduction, begins with a general hypothesis and uses definitions, logic, and established results to show that a certain conclusion must be true. This is the type of reasoning used in most direct proofs. Deductive reasoning moves from the general to the specific. With inductive reasoning, a conclusion is reached based on a set of observations. Patterns are established and conclusions are drawn based on those patterns. Inductive reasoning moves from the specific to the general. EX: Determine the type of reasoning demonstrated in each example. ..The high temperatures over the past 7 days were 77°, 78°, 79°, 80°, 81°, 82°, and 83°. Each day the temperature was 1° warmer than the previous day. So it can be concluded that the high temperature the next day will be 84°. ..All squares are rectangles, and all rectangles are parallelograms. ABCD is a square, so ABCD is a rectangle. Since ABCD is a rectangle, ABCD is also a parallelogram. So it can be concluded that ABCD is a parallelogram. ..In the first example, a pattern is observed and a conclusion is drawn based on the pattern of observations. This is an example of inductive reasoning. ..In the second example, a hypothesis is given and a series of logical statements is given to reach a conclusion about a specific quadrilateral. This is an example of deductive reasoning.

Proof by Mathematical Induction Inductive reasoning is not a valid method of proof, but it forms the basis of a method called mathematical induction. Mathematical induction can be used to prove statements P(n) that are defined for integers n. Mathematical Induction Steps To prove a statement P(n) is true for integers n: 1. Show that P(a) is true for a specific integer a. This is called the basis step of the proof. 2. Suppose that P(k) is true for a general integer k ≥ a. Then show that P(k + 1) must also be true. This is called the inductive step of the proof. n

Π (k + 1) = (1 + 1)(2 + 1)(3 + 1)(4 + 1) = 2 ∙ 3 ∙ 4 ∙ 5 = 120

EX: Prove:

k =1

Factorial Notation

The product of all integers from 1 to n is called n factorial and is expressed as n! = n ∙ (n – 1) ∙ . . . ∙ 3 ∙ 2 ∙ 1. EX: Simplify each expression. 7! = 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 5,040

10! 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = = 10 ∙ 9 ∙ 8 = 720 7! 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 Recursion While some sequences are generated by an explicit formula, others are defined recursively. The terms of a sequence defined by a recursive formula are found using the previous terms of the sequence. EX: Find the first 5 terms of the sequence in which a1 = –3 and an + 1 = –2an + 1. Substitute the previous term into the recursive formula to find each subsequent term. a1 = –3 a2 = –2a1 + 1 = –2(–3) + 1 = 7 a3 = –2a2 + 1 = –2(7) + 1 = –13 a4 = –2a3 + 1 = –2(–13) + 1 = 27 a5 = –2a4 + 1 = –2(27) + 1 = –53 The first 5 terms are –3, 7, –13, 27, and –53. EX: The Fibonacci Sequence is a recursively defined sequence that often occurs in nature. It is defined as a1 = 1, a2 = 1, and an = an – 2 + an – 1. Find the first 8 terms of the Fibonacci Sequence. a1 = 1 a2 = 1 a3 = 1 + 1 = 2 a4 = 1 + 2 = 3 a5 = 2 + 3 = 5 a6 = 3 + 5 = 8 a7 = 5 + 8 = 13 a8 = 8 + 13 = 21 The first 8 terms are 1, 1, 2, 3, 5, 8, 13, and 21. 4

∑ k = 1 + 2 + 3 + … + n = n (n + 1) . k =1

n (n + 1) Let P(n) = 1 + 2 + 3 + … + n = . 2 1. Show that the P(n) is true for n = 1. P(1) =

2

1(1+ 1) 2 = =1 2 2

2. Suppose P(n) is true for some integer k ≥ 1. That is, assume that 1 + 2 + 3 + … + k =

k (k + 1) is true. 2

Now show that P(k + 1) must also be true. Begin with the assumed statement. 1+2+3+…+k=

k (k + 1) 2

Add k + 1 to each side. k (k + 1) 1 + 2 + 3 + … + k + (k + 1) = + (k + 1)

2

Simplify the right hand side. 1 + 2 + 3 + … + k + (k + 1) =

k (k + 1) 2(k + 1) + 2 2

1 + 2 + 3 + …+ k + (k + 1) =

k (k + 1) + 2(k + 1) 2

Notice that the final statement above is P(k + 1). So it has been shown that if P(k) is true, then P(k + 1) must be true. n

n (n + 1) Therefore, ∑ k = 1 + 2 + 3 + … + n = . k =1

2

Sets ••Distributive Property A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) ••Complement of a Complement Property (Ac)c = A ••De Morgan’s Laws for Sets (A ∩ B)c = Ac ∪ Bc (A ∪ B)c = Ac ∩ Bc

Definitions

A set is a collection of objects (such as numbers). Each member is called an element of the set. If A and B are sets, A is a subset of B if and only if every element of A is also an element of B. This is written A ⊆ B which is read “A is a subset of B” or “B contains A.” If C and D are sets, C is a proper subset of D if and only if every element of C is an element of D, but there is at least one element of D that is not in C. EX: ..The set N = {red, blue, green, yellow, purple, orange} has 6 elements. ..The set M = {red, green, yellow, orange} is a proper subset of N. Every element of M is an element of N, but there is at least one element in N that is not in M.

A Venn diagram is a visual representation that shows the relationship between sets. EX: The set of whole numbers W = {0, 1, 2, 3, …} is a proper subset of the set of integers Z = {… , –3, –2, –1, 0, 1, 2, 3, …}.

W

Ordered n-tuples & Cartesian Products An ordered n-tuple is an arrangement of n elements of a set in which the order is important. The notation for an ordered n-tuple is (x1, x2, … , xn) in which x1 is first, x2 is second, and so on. An ordered 2-tuple is also called an ordered pair (x, y), and an ordered 3-tuple is called an ordered triple (x, y, z). Two ordered n-tuples (x1, x2, … , xn) and (y1, y2, … , yn) are equal if and only if x1 = y1, x2 = y2, … , xn = yn. For two sets A and B, the Cartesian product of A and B is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B. The notation for a Cartesian product is A × B which is read “A cross B.”

Z

Unions, Intersections & Complements

In mathematics, a universal set, U, is the set of all elements under consideration for a particular situation. The universal set could be the set of all integers, the set of all real numbers, the set of all positive integers, etc. For a universal set U and sets A and B with A ⊆ U and B ⊆ U , the following definitions hold. ••The union of A and B is the set of all elements x in U such that x ∈ A or x ∈ B. The notation is A ∪ B. ••The intersection of A and B is the set of all elements x in U such that x ∈ A and x ∈ B. The notation is A ∩ B. ••The complement of A is the set of all elements x in U such that x ∉ A. The notation is Ac.

EX: Given the sets A = {1, 2} and B = {3, 4, 5}, find A × B. A × B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}

The Empty Set

The empty set (or null set) is a unique set that contains no elements. The empty set is represented symbolically as ∅ or {}. There are several properties of sets that involve the empty set. For a universal set U and all sets A, the following properties are true: ••A ∪ ∅ = A ••A ∩ Ac = ∅ ••A ∪ Ac = U ••A ∩ ∅ = ∅ ••U c = ∅ ••∅ c = U

EX: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, R = {2, 4, 6, 8, 10}, and S = {1, 2, 3, 4, 5}. R ∪ S = {1, 2, 3, 4, 5, 6, 8, 10} R ∩ S = {2, 4} Rc = {1, 3, 5, 7, 9} Sc = {6, 7, 8, 9, 10}

Properties of Sets

For subsets A, B, and C of a universal set U, the following identities are true. ••Associative Property (A ∩ B) ∩ C = A ∩ (B ∩ C) (A ∪ B) ∪ C = A ∪ (B ∪ C) ••Commutative Property A∩B=B∩A A∪B=B∪A

Partitions of Sets Two sets are disjoint if the sets have no elements in common. A partition of a set A is a collection of nonempty sets {A1, A2, … , An} such that A = A1 ∪ A2 ∪…∪ An and A1, A2, … , An are mutually disjoint. EX: Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A partition of set A is {A1, A2, A3}, where A1 = {1, 2}, A2 = {3, 4, 5, 6, 7}, and A3 = {8, 9, 10}.

Counting An experiment or process is random if it is not possible to predict the outcome with certainty for each trial. It is often useful to be able to count all of the possible outcomes of a random process or the subsets of a given set.

The Fundamental Counting Principle The Fundamental Counting Principle says that if a process A has a possible outcomes, process B has b possible outcomes, process C has c possible outcomes, . . . , and process N has n possible outcomes, then the total number of ways all of the processes can occur is a ∙ b ∙ c ∙ … ∙ n. EX: A deli offers 5 different kinds of bread, 8 different kinds of sandwiches, and 7 different side dishes. How many possible lunches are there if a customer selects a kind of bread, type of sandwich, and side dish at random? There are 5 ∙ 8 ∙ 7 = 280 possible lunches.

Combinations

A combination is a set of objects in which the order of the objects is not important. EX: Let U = {Bryan, Tim, Ling, Ana, Grace}. The subset {Tim, Ana} is a combination of 2 elements from U. The subset {Ana, Tim} represents the same combination because the order of the elements is not important.

The number of combinations of n objects taken r n! C = at a time is given by the formula n r (n − r )! r ! . The expression nCr is read “n choose r,” and the expression 6C3 is read “6 choose 3.”

EX: There are 12 books on a shelf. Meredith selects 2 of the books at random to take on vacation. How many possible combinations of books are there? 12! 12! n = 12, r = 2 12 C2 = (12 − 2)!2! = 10!2! = 66

Permutations

A permutation is a set of objects in which the order of the objects is important. A set that contains n elements has n! different permutations. This is because there are n possible outcomes for the first choice, (n – 1) possible outcomes for the second choice, and so on. EX: How many permutations are there of the elements of the set {red, blue, green}? There are 3! = 6 permutations as shown below: red, blue, green red, green, blue blue, red, green blue, green, red green, red, blue green, blue, red

The number of permutations of n objects taken r at n!

a time is given by the formula n Pr = (n − r )! . EX: There are 8 swimmers in a race. In how many ways can first place, second place, and third place be awarded?

8! 8! n = 8, r = 3 8 P3 = (8 − 3)! = 5! = 336 5

Counting & Probability

A sample space is the set of all possible outcomes for a random experiment or process. An event is a subset of a sample space of outcomes. The probability of an event is a numerical value that describes how likely the event is to occur. Probability ranges from 0 (impossible) to 1 (certain). It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space. The probability of an event E is denoted P(E). The counting methods described earlier play an important role in calculating probability. P(event) =

number of favorable outcomes total number of outcomes

EX: There are 6 sophomores, 8 juniors, and 10 seniors on a committee. A subcommittee of 6 students is chosen at random. What is the probability that the subcommittee contains 2 sophomores, 2 juniors, and 2 seniors? Find the total number of outcomes (combinations) when 6 students are chosen from a group of 24. 24

C6 =

24! 24! = = 134,596 (24 − 6)!6! 18!6!

Use the Fundamental Counting Principle to find the number of ways 2 sophomores, 2 juniors, and 2 seniors can be chosen. (6C2)(8C2)(10C2) = 18,900 P(2 sophomores, 2 juniors, 2 seniors) =

18,900 134,596 ≈ 0.14

Functions & Set Relations Definitions

Injective, Surjective & Bijective Functions

A relation R from set A to set B is any subset of the Cartesian product A × B. For a given ordered pair (x, y) in A × B, x R y (read “x is related to y by R”) if and only if (x, y) ∈ R.

A function f: A → B is injective, or one-to-one, if and only if each element of the codomain is mapped by at most one element of the domain. In other words, if x1 and x2 and elements of the domain where x1 ≠ x2, then f(x1) ≠ f(x2).

EX: Let A = {1, 2, 3} and B = {4, 5}. Define the relation R to be any (x, y) in A × B such that |y – x| is odd. Determine which ordered pairs are in R. A × B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} Of these ordered pairs, only (1, 4), (2, 5), and (3, 4) satisfy the rule |y – x| is odd. So, R = {(1, 4), (2, 5), (3, 4)}.

EX: The function g: C → D shown in the mapping diagram is injective because no two elements of the domain are mapped to the same element of the codomain.

A function f: A → B is surjective, or onto, if and only if for each element of the codomain there is at least one element in the domain that maps to it. In other words, if y1 is any element of the codomain B, then there exists an element x1 in A such that f(x1) = y1. For a surjective function, the range is equivalent to the codomain.

A function f from set A to set B is a relation between the elements of A and the elements of B such that each element of A is related to a unique element of B. The notation f: A → B is read “f is a function from A to B.” A is the domain, or the set of possible input values for the function. B is the codomain, or the set of possible output values for the function. The range of a function is the actual set of output values from the codomain that are paired with input values from the domain. A mapping diagram uses arrows to show how elements from the domain are paired with elements from the codomain.

EX: The function h: M → N shown in the mapping diagram is surjective because each element of the codomain has an element of the domain that maps to it.

EX: The mapping diagram shows a function f: A B f A → B. Identify the domain, codomain, and range of the function. Then list the ordered –2 1 pairs in f. 3 8 3 7 The domain is the set of possible input values, –4 0 or {–2, 3, 7, 0}. –6 The codomain is the set of possible output values, or {1, 8, 3, –4, –6}. The range is the set of output values that are paired with input values from the domain, or {1, 3, –6}. f = {(–2, 3), (3, 1), (7, –6), (0, –6)}

A function f: A → B that is both injective and surjective is a bijective function. Another way of saying that a function is bijective is to say that there is a one-to-one correspondence from A to B.

C

g

a b c d

M 6 3 1 9 0

D m n o p q

h

N –2 –1 –7 –9

Graphs & Trees Graphs

A graph is defined by a finite set of points, called vertices, and a finite set of line segments, called edges, that connect the vertices. Each edge is associated with either one or two vertices. An edge that is associated with just one vertex is called a loop. Two distinct edges that are associated with the same set of vertices are parallel edges. A simple graph is a graph that does not have any loops or parallel edges. v2 e1 v1

e3 e2

v3

e4

v4

e5 v5

e6

EX: Describe the edges and vertices of the graph. • The vertices of the graph are {v1, v2, v3, v4, v5}. • The edges of the graph are {e1, e2, e3, e4, e5, e6}. • Edges e1 and e2 are parallel because they are distinct edges that are both associated with vertices v1 and v2. • Edge e6 is a loop because it is associated with only one vertex, v4.

Walks, Paths & Circuits

For a given graph with vertices v1 and v2, a walk from v1 to v2 is an alternating sequence of adjacent vertices and edges. EX: In the preceding graph, one walk from v1 to v5 would be v1e2v2e3v3e5v5.

A path from v1 to v2 is a walk that does not contain any repeated edges. A simple path is a path that does not contain any repeated vertices. The walk described in

the previous example is a simple path because none of the edges or vertices in the walk repeat. A closed walk is a walk that begins and ends at the same vertex. A circuit is a closed walk that does not contain a repeated edge. A simple circuit is a circuit with no repeated vertices except for the beginning and ending vertex. EX: For this graph shown, describe each of the following walks as a path, a simple path, a circuit, or a simple circuit. e6 • v5e8v4e6v3e3v2e1v1 is a v3 e2 simple path. It is a walk v1 e7 v4 from v5 to v1 and has no repeated vertices or e1 e5 e 8 e3 edges. • v1e2v3e6v4e7v3e3v2e1v1 is v5 a circuit. It is a closed v2 e4 walk from v1 to v1 with no e9 repeated edges, but it does have repeated vertices (v3). • v1e2v3e6v4e7v3e5v5 is a path. It is a walk from v1 to v5 with no repeated edges, but it does have repeated vertices (v3). • v2e3v3e5v5e4v2 is a simple circuit. It is a closed walk from v2 to v2 with no repeated edges or vertices other than the beginning and ending vertex.

Trees

Two vertices of a graph, v1 and v2, are connected if and only if there is a walk from v1 to v2. A graph is connected if and only if every pair of vertices in the graph is connected. A trivial circuit is a circuit that consists of only a single vertex. A graph is a tree if and only if it is connected and has only trivial circuits.

EX: The graphs shown below are trees. Every pair of vertices is connected and the only circuits are trivial circuits.

Decision trees are used in a variety of situations to make decisions based on a number of factors. EX: The decision tree below could be used to determine whether a hot air balloon company flies depending on the wind speeds and visibility conditions. >5 miles 5 miles Visability Conditions