Discrete Mathematics with Application-4th Edition by Susanna S. Epp PDF

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This page was intentionally left blank List of Symbols Subject Symbol Meaning Page Logic ∼p not p 25 p∧q p and q 25 p∨q p or q 25 p ⊕ q or p XOR q p or q but not both p and q 28 P≡Q P is logically equivalent to Q 30 p→q if p then q 40 p↔q p if and only if q 45 ∴ therefore 51 P(x) predicate in x 97 ...

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List of Symbols Subject

Symbol

Meaning

Logic

∼p

not p

25

p and q

25

p∨q

p or q

25

p or q but not both p and q

28

P≡Q

P is logically equivalent to Q

30

p∧q

p ⊕ q or p XOR q p→q

if p then q

40

p if and only if q

45

∴

therefore

51

P(x)

predicate in x

P(x) ⇒ Q(x)

every element in the truth set for P(x) is in the truth set for Q(x)

p↔q

Number Theory and Applications

97 104

P(x) and Q(x) have identical truth sets

104

for all

101

there exists

103

NOT-gate

67

AND

AND-gate

67

OR

OR-gate

67

NAND

NAND-gate

75

NOR

NOR-gate

75

P(x) ⇔ Q(x)

∀

Applications of Logic

Page

∃

NOT

|

Sheffer stroke

74

↓

Peirce arrow

74

n2

number written in binary notation

78

n 10

number written in decimal notation

78

n 16

number written in hexadecimal notation

91

d divides n

170

d does not divide n

172

n div d

the integer quotient of n divided by d

181

n mod d

the integer remainder of n divided by d

181

⌊x⌋

the floor of x

191

d |n

d /| n

the ceiling of x

191

|x|

the absolute value of x

187

gcd(a, b)

the greatest common divisor of a and b

220

x := e

x is assigned the value e

214

⌈x⌉

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Subject

Symbol

Meaning

Page

Sequences

... n 

and so forth

227

ak

the summation from k equals m to n of ak

230

ak

the product from k equals m to n of ak

223

n!

n factorial

237

a∈A

a is an element of A

k=m n 

k=m

Set Theory

7

a is not an element of A

7

{a1 , a2 , . . . , an }

the set with elements a1 , a2 , . . . , an

7

R, R− , R+ , Rnonneg

the sets of all real numbers, negative real numbers, positive real numbers, and nonnegative real numbers

7, 8

Z, Z− , Z+ , Znonneg

the sets of all integers, negative integers, positive integers, and nonnegative integers

7, 8

Q, Q− , Q+ , Qnonneg

the sets of all rational numbers, negative rational numbers, positive rational numbers, and nonnegative rational numbers

7, 8

N

the set of natural numbers

8

A⊆B

A is a subset of B

9

A is not a subset of B

9

a∈ / A

{x ∈ D | P(x)}

A ⊆ B

A=B

A∪B

the set of all x in D for which P(x) is true

8

A equals B

339

A union B

341

A intersect B

341

B−A

the difference of B minus A

341

Ac

the complement of A

341

A∩B

(x, y)

ordered pair

(x 1 , x2 , . . . , xn )

ordered n-tuple

A×B

the Cartesian product of A and B the Cartesian product of A1 , A2 , . . . , An

347

∅

the empty set

361

the power set of A

346

A1 × A2 × · · · × An

P( A)

11 346 12

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List of Symbols Subject

Symbol

Meaning

Counting and Probability

N (A)

the number of elements in set A

518

P( A)

the probability of a set A

518

P(n, r )

the number of r -permutations of a set of n elements

553

n 

n choose r , the number of r -combinations of a set of n elements, the number of r -element subsets of a set of n elements

566

[xi1 , xi2 , . . . , xir ]

multiset of size r

584

the probability of A given B

612

r

P( A | B)

Functions

f: X → Y f (x) f

x →y

Relations

f is a function from X to Y

384

the value of f at x

384

f sends x to y

384

f (A)

the image of A

397

f −1 (C)

the inverse image of C

397

Ix

the identity function on X

387

x

b raised to the power x

405, 406

expb (x)

b raised to the power x

405, 406

logb (x)

logarithm with base b of x

388

F −1

the inverse function of F

411

f ◦g

the composition of g and f

417

b

Algorithm Efficiency

Page

x∼ =y

x is approximately equal to y

237

O( f (x))

big-O of f of x

727

( f (x))

big-Omega of f of x

727

( f (x))

big-Theta of f of x

727

xRy

x is related to y by R

R

−1

m ≡ n (mod d) [a]

xy

the inverse relation of R

14 444

m is congruent to n modulo d

473

the equivalence class of a

465

x is related to y by a partial order relation 

502

Continued on first page of back endpapers.

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DISCRETE MATHEMATICS

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DISCRETE MATHEMATICS WITH APPLICATIONS FOURTH EDITION

SUSANNA S. EPP DePaul University

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Cover Photo: The stones are discrete objects placed one on top of another like a chain of careful reasoning. A person who decides to build such a tower aspires to the heights and enjoys playing with a challenging problem. Choosing the stones takes both a scientific and an aesthetic sense. Getting them to balance requires patient effort and careful thought. And the tower that results is beautiful. A perfect metaphor for discrete mathematics! Discrete Mathematics with Applications, Fourth Edition Susanna S. Epp Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Associate Editor: Daniel Seibert Editorial Assistant: Shaylin Walsh Associate Media Editor: Andrew Coppola Senior Marketing Manager: Jennifer Pursley Jones Marketing Communications Manager: Mary Anne Payumo Marketing Coordinator: Erica O’Connell

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To Jayne and Ernest

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CONTENTS Chapter 1 Speaking Mathematically 1.1 Variables

1

1

Using Variables in Mathematical Discourse; Introduction to Universal, Existential, and Conditional Statements

1.2 The Language of Sets

6

The Set-Roster and Set-Builder Notations; Subsets; Cartesian Products

1.3 The Language of Relations and Functions

13

Definition of a Relation from One Set to Another; Arrow Diagram of a Relation; Definition of Function; Function Machines; Equality of Functions

Chapter 2 The Logic of Compound Statements

23

2.1 Logical Form and Logical Equivalence

23

Statements; Compound Statements; Truth Values; Evaluating the Truth of More General Compound Statements; Logical Equivalence; Tautologies and Contradictions; Summary of Logical Equivalences

2.2 Conditional Statements

39

Logical Equivalences Involving →; Representation of If-Then As Or; The Negation of a Conditional Statement; The Contrapositive of a Conditional Statement; The Converse and Inverse of a Conditional Statement; Only If and the Biconditional; Necessary and Sufficient Conditions; Remarks

2.3 Valid and Invalid Arguments

51

Modus Ponens and Modus Tollens; Additional Valid Argument Forms: Rules of Inference; Fallacies; Contradictions and Valid Arguments; Summary of Rules of Inference

2.4 Application: Digital Logic Circuits

64

Black Boxes and Gates; The Input/Output Table for a Circuit; The Boolean Expression Corresponding to a Circuit; The Circuit Corresponding to a Boolean Expression; Finding a Circuit That Corresponds to a Given Input/Output Table; Simplifying Combinational Circuits; NAND and NOR Gates

2.5 Application: Number Systems and Circuits for Addition

78

Binary Representation of Numbers; Binary Addition and Subtraction; Circuits for Computer Addition; Two’s Complements and the Computer Representation of vi

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Contents

vii

Negative Integers; 8-Bit Representation of a Number; Computer Addition with Negative Integers; Hexadecimal Notation

Chapter 3 The Logic of Quantified Statements

96

3.1 Predicates and Quantified Statements I

96

The Universal Quantifier: ∀; The Existential Quantifier: ∃; Formal Versus Informal Language; Universal Conditional Statements; Equivalent Forms of Universal and Existential Statements; Implicit Quantification; Tarski’s World

3.2 Predicates and Quantified Statements II

108

Negations of Quantified Statements; Negations of Universal Conditional Statements; The Relation among ∀, ∃, ∧, and ∨; Vacuous Truth of Universal Statements; Variants of Universal Conditional Statements; Necessary and Sufficient Conditions, Only If

3.3 Statements with Multiple Quantifiers

117

Translating from Informal to Formal Language; Ambiguous Language; Negations of Multiply-Quantified Statements; Order of Quantifiers; Formal Logical Notation; Prolog

3.4 Arguments with Quantified Statements

132

Universal Modus Ponens; Use of Universal Modus Ponens in a Proof; Universal Modus Tollens; Proving Validity of Arguments with Quantified Statements; Using Diagrams to Test for Validity; Creating Additional Forms of Argument; Remark on the Converse and Inverse Errors

Chapter 4 Elementary Number Theory and Methods of Proof

145

4.1 Direct Proof and Counterexample I: Introduction

146

Definitions; Proving Existential Statements; Disproving Universal Statements by Counterexample; Proving Universal Statements; Directions for Writing Proofs of Universal Statements; Variations among Proofs; Common Mistakes; Getting Proofs Started; Showing That an Existential Statement Is False; Conjecture, Proof, and Disproof

4.2 Direct Proof and Counterexample II: Rational Numbers

163

More on Generalizing from the Generic Particular; Proving Properties of Rational Numbers; Deriving New Mathematics from Old

4.3 Direct Proof and Counterexample III: Divisibility

170

Proving Properties of Divisibility; Counterexamples and Divisibility; The Unique Factorization of Integers Theorem

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