Discrete Mathematics with Applications

932 Pages·2003·47.76 MB·English

by  Susanna S.(Susanna S. Epp) Epp

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List of Symbols
Subject Symbol Meaning Page
Logic not p 3
pAq pandq 3
pvq porq 3
p E q or p XOR q p or q but not both p and q 7
P Q P is logically equivalent to Q 8
p q if pthenq 18
p q p if and only if q 24
therefore 29
P(x) predicate in x 76
P(x) =t- Q(x) every element in the truth set for P( x) is in 84
the truth set for Q(x)
P(x) <> Q (x P(x) and Q(x) have identical truth sets 84
V for all 78
A there exists 79
Applications
of Logic N NOT-gate 46
AND-gate 46
ROR-gate 46
NAND NAND-gate 54
NOR-gate 54
Sheffer stroke 54
4 Peirce arrow 54
r2 number written in binary notation 58
nOo number written in decimal notation 58
n16 number written in hexadecimal notation 71
Number d In d divides n 148
Theory and d% n d does not divide n 149
Applications n div d the integer quotient of n divided by d 158
n mod d the remainder of n divided by d 158
[xj the floor of x 165
Fxl the ceiling of x 165
W the absolute value of x 164
gcd(a, b) the greatest common divisor of a and b 192
x := e x is assigned the value e 186
Subject Symbol Meaning Page
Sequences ... and so forth 199
n
E ak the summation from k equals m to n of ak 202
k=m
n
fl -k the product from k equals m to n of a, 205
k=m
n& n factorial 206
Set a E A a is an element of A 76
Theory a 0 A a is not an element of A 76
(a,, a2, ... , a.) the set with elements a,, a2, ..., an 76
Ix E D I Plx)} the set of all x in D for which P(x) is true 77
R, R-, R+, Ronneg the sets of all real numbers, negative real 76, 77
numbers, positive real numbers, and
nonnegative real numbers
Z' Z-' Z+ znonneg the sets of all integers, negative integers, 76, 77
positive integers, and nonnegative integers
Q, Q- Q+, Qnonneg the sets of all rational numbers, negative 76, 77
rational numbers, positive rational numbers,
and nonnegative rational numbers
N the set of natural numbers 77
A C B A is a subset of B 256
A g B A is not a subset of B 257
A=B
A =B A equals B 258
A U B A union B 260
A n B A intersect B 260
the difference of B minus A 260
A' the complement of A 260
(xby) ordered pair 264
ordered n-tuple 264
(XI, X29 * * * , Xn)
A x B the Cartesian product of A and B 265
A, x A2 x.. X An the Cartesian product of Al, A2 . A, 265
0 the empty set 262
97(A) the power set of A 264
List of Symbols
Subject Symbol Meaning Page
Counting and N (A) the number of elements in a set A 299
Probability P(A) the probability of a set A 299
P(n, r) the number of r-permutations of a set of 315
n elements
(n) n choose r, the number of r-combinations 334
r of a set of n elements, the number of
r-element subsets of a set of n elements
[x,1, Xi2 .... X i I multiset of size r 349
P(A IB ) the probability of A given B 376
Functions f: X -+ Y f is a function from X to Y 390
f (x) the value of f at x 390
x-f|y f sends x to y 390
f (A) the image of A 402
f 1( C) the inverse image of C 402
ix the identity function on X 394
bx b raised to the power x 411
expb (x) b raised to the power x 411
1ogb (x) logarithm with base b of x 395
F-1 the inverse function of F 415
f og the composition of g and f 432
Algorithm x - y x is approximately equal to y 206
Efficiency O(f( x)) big-0 of f of x 519
(f (x)) big-Omega of f of x 519
J(f( x)) big-Theta of f of x 519
Relations x R y x is related to y by R 572
R the inverse relation of R 578
m = n (mod d) m is congruent to n modulo d 597
[a] the equivalence class of a 599
x -< y x is related to y by a partial order relation < 635
Continued on first page of back endpapers.
DISCRETE MATHEMATICS
WITH APPLICATIONS
THIRD EDITION
SUSANNI k S. EPP
DePaul Uuniversity
TF-IcHIVIScON
BROOKS/COLE
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United Kingdom * United States
THCOlVISCON
BROOKS/COLE
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 metaphorf or discrete mathematics!
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CONTENTS
Chapter 1 The Logic of Compound Statements 1
1.1 Logical Form and Logical Equivalence 1
Statements; Compound Statements; Truth Values; Evaluating the Truth of More Gen-
eral Compound Statements; Logical Equivalence; Tautologies and Contradictions;
Summary of Logical Equivalences
1.2 Conditional Statements 17
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
1.3 Valid and Invalid Arguments 29
Modus Ponens and Modus Tollens; Additional Valid Argument Forms: Rules of
Inference; Fallacies; Contradictions and Valid Arguments; Summary of Rules of
Inference
1.4 Application: Digital Logic Circuits 43
Black Boxes and Gates; The Input/Output for a Circuit; The Boolean Expression Cor-
responding 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
1.5 Application: Number Systems and Circuits for Addition 57
Binary Representation of Numbers; Binary Addition and Subtraction; Circuits for
Computer Addition; Two's Complements and the Computer Representation of Neg-
ative Integers; 8-Bit Representation of a Number; Computer Addition with Negative
Integers; Hexadecimal Notation
Chapter 2 The Logic of Quantified Statements 75
2.1 Introduction to Predicates and Quantified Statements I 75
The Universal Quantifier: V; The Existential Quantifier: B; Formal Versus Informal
Language; Universal Conditional Statements; Equivalent Forms of the Universal and
Existential Statements; Implicit Quantification; Tarski's World
2.2 Introduction to Predicates and Quantified Statements II 88
Negations of Quantified Statements; Negations of Universal Conditional Statements;
The Relation among V, 3, A, and v; Vacuous Truth of Universal Statements; Variants
of Universal Conditional Statements; Necessary and Sufficient Conditions, Only If
iv
Contents V
2.3 Statements Containing Multiple Quantifiers 97
Translating from Informal to Formal Language; Ambiguous Language; Negations
of Multiply-Quantified Statements; Order of Quantifiers; Formal Logical Notation;
Prolog
2.4 Arguments with Quantified Statements 111
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 3 Elementary Number Theory
and Methods of Proof 125
3.1 Direct Proof and Counterexample I: Introduction 126
Definitions; Proving Existential Statements; Disproving Universal Statements by
Counterexample; Proving Universal Statements; Directions for Writing Proofs of
Universal Statements; Common Mistakes; Getting Proofs Started; Showing That an
Existential Statement Is False; Conjecture, Proof, and Disproof
3.2 Direct Proof and Counterexample II: Rational Numbers 141
More on Generalizing from the Generic Particular; Proving Properties of Rational
Numbers; Deriving New Mathematics from Old
3.3 Direct Proof and Counterexample Ill: Divisibility 148
Proving Properties of Divisibility; Counterexamples and Divisibility; The Unique
Factorization Theorem
3.4 Direct Proof and Counterexample IV, Division into Cases
and the Quotient-Remainder Theorem 156
Discussion of the Quotient-Remainder Theorem and Examples; div and mod; Alter-
native Representations of Integers and Applications to Number Theory
3.5 Direct Proof and Counterexample V Floor and Ceiling 164
Definition and Basic Properties; The Floor of n/2
3.6 Indirect Argument: Contradiction and Contraposition 171
Proof by Contradiction; Argument by Contraposition; Relation between Proof by
Contradiction and Proof by Contraposition; Proof as a Problem-Solving Tool
3.7 Two Classical Theorems 179
The Irrationality of vf2; The Infinitude of the Set of Prime Numbers; When to Use
Indirect Proof; Open Questions in Number Theory
vi Contents
3.8 Application: Algorithms 186
An Algorithmic Language; A Notation for Algorithms; Trace Tables; The Division
Algorithm; The Euclidean Algorithm
Chapter 4 Sequences and Mathematical Induction 199
4.1 Sequences 199
Explicit Formulas for Sequences; Summation Notation; Product Notation; Factorial
Notation; Properties of Summations and Products; Change of Variable; Sequences in
Computer Programming; Application: Algorithm to Convert from Base 10 to Base 2
Using Repeated Division by 2
4.2 Mathematical Induction I 215
Principle of Mathematical Induction; Sum of the First n Integers; Sum of a Geometric
Sequence
4.3 Mathematical Induction II 227
Comparison of Mathematical Induction and Inductive Reasoning; Proving Divisibility
Properties; Proving Inequalities
4.4 Strong Mathematical Induction
and the Well-Ordering Principle 235
The Principle of Strong Mathematical Induction; Binary Representation of Integers;
The Well-Ordering Principle for the Integers
4.5 Application: Correctness of Algorithms 244
Assertions; Loop Invariants; Correctness of the Division Algorithm; Correctness of
the Euclidean Algorithm
Chapter 5 Set Theory 255
5.1 Basic Definitions of Set Theory 255
Subsets; Set Equality; Operations on Sets; Venn Diagrams; The Empty Set; Partitions
of Sets; Power Sets; Cartesian Products; An Algorithm to Check Whether One Set Is
a Subset of Another (Optional)
5.2 Properties of Sets 269
Set Identities; Proving Set Identities; Proving That a Set Is the Empty Set
5.3 Disproofs, Algebraic Proofs, and Boolean Algebras 282
Disproving an Alleged Set Property; Problem-Solving Strategy; The Number of Sub-
sets of a Set; "Algebraic" Proofs of Set Identities; Boolean Algebras