Table Of ContentDiscrete Mathematices
witLh Graph Thoery
Second Edition
Edgar G. Goodoire * Michael M. Pormenlte r
Notation
Here, and on the last two end papers, is a list of the symbols and other notation used in this book grouped, as
best as possible, by subject. Page numbers give the location of first appearance.
Page Symbol Meaning == rr
Miscellaneous
5 / used to express the negation of any symbol over which it is written;
for example, , means "does not belong to"
12 I used to denote the end of a proof
76 jx the absolute value of x
90 t1% pronounced "aleph naught," this is the cardinality of the natural
numbers
121 approximately
>1
152 sum
162 11 product
Logical
3 implies
4 if and only if
5 negation
7 V for all
7 3 there exists
17 A and
17 V or
22 denotes logical equivalence
22 contradiction
22 1 tautology
Common Sets
37 N or N the natural numbers
38 Q or Q the rational numbers
38 R or JR the real numbers
38 Z or Z the integers
39 C or C the complex numbers
81 R+ the positive real numbers
-1
Discrete
Mathematics
with Graph Theory
Discrete
Mathematics
with Graph Theory
Second Edition
Edgar G. Goodaire
Memorial University of Newfoundland
Michael M. Parmenter
Memorial University of Newfoundland
PRENTICE HALL
Upper Saddle River, NJ 07458
Library of Congress Cataloging-in-Publication Data
Goodaire, Edgar G.
Discrete mathematics with graph theory / Edgar G. Goodaire, Michael M. Parmenter.-
nd ed.
2
p. cm.
Includes bibliographical references and index.
ISBN 0-13-092000-2
1. Mathematics. 2. Computer science-Mathematics. 3. Graph theory. I. Parmenter,
Michael M. II. Title.
QA39.3.G66 2002
511 -dc2l 2001037448
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©)2 002, 1998 by Prentice-Hall, Inc.
Upper Saddle River, NJ 07458
All rights reserved. No part of this book may be
reproduced, in any form or by any means,
without permission in writing from the publisher.
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
ISBN 0-13-092000-2
Pearson Education Ltd., London
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To Linda E. G. G.
To Brenda M. M. P.
Only those who live with an author can appreciate the work that goes
into writing. We are sincerely grateful for the loving encouragement
and patience of our wives over a period of years while we have worked
on two editions of this book.
Contents
Preface xi
Suggested Lecture Schedule xvii
Yes, There are Proofs! 1
1.1 Compound Statements 2
1.2 Proofs in Mathematics 9
1.3 Truth Tables 17
1.4 The Algebra of Propositions 21
1.5 Logical Arguments 28
Review Exercises 34
2
Sets and Relations 37
2.1 Sets 37
2.2 Operations on Sets 43
2.3 Binary Relations 51
2.4 Equivalence Relations 56
2.5 Partial Orders 63
Review Exercises 68
3
Functions 71
3.1 Domain, Range, One-to-One, Onto 71
3.2 Inverses and Composition 79
3.3 One-to-One Correspondence and the Cardinality of a Set 87
Review Exercises 95
4
The Integers 97
4.1 The Division Algorithm 97
4.2 Divisibility and the Euclidean Algorithm 104
4.3 Prime Numbers 114
4.4 Congruence 126
4.5 Applications of Congruence 136
Review Exercises 147
vii
Viii Contents
5
Induction and Recursion 149
5.1 Mathematical Induction 149
5.2 Recursively Defined Sequences 163
5.3 Solving Recurrence Relations; The Characteristic Polynomial 173
5.4 Solving Recurrence Relations; Generating Functions 178
Review Exercises 185
6
Principles of Counting 187
6.1 The Principle of Inclusion-Exclusion 187
6.2 The Addition and Multiplication Rules 196
6.3 The Pigeon-Hole Principle 204
Review Exercises 209
7
Permutations and Combinations 211
7.1 Permutations 211
7.2 Combinations 216
7.3 Repetitions 223
7.4 Derangements 228
7.5 The Binomial Theorem 231
Review Exercises 237
8
Algorithms 239
8.1 What Is an Algorithm? 239
8.2 Complexity 246
8.3 Searching and Sorting 259
8.4 Enumeration of Permutations and Combinations 271
Review Exercises 275
9
Graphs 277
9.1 A Gentle Introduction 277
9.2 Definitions and Basic Properties 286
9.3 Isomorphism 294
Review Exercises 299
Q
1
Paths and Circuits 303
10.1 Eulerian Circuits 303
10.2 Hamiltonian Cycles 310
10.3 The Adjacency Matrix 318
10.4 Shortest Path Algorithms 325
Review Exercises 333
Contents iX
I I
Applications of Paths and Circuits 337
11.1 The Chinese Postman Problem 337
11.2 Digraphs 342
11.3 RNA Chains 350
11.4 Tournaments 355
11.5 Scheduling Problems 360
Review Exercises 365
1 2
Trees 367
12.1 What Is a Tree? 367
12.2 Properties of Trees 372
12.3 Spanning Trees 377
12.4 Minimum Spanning Tree Algorithms 382
12.5 Acyclic Digraphs and Bellman's Algorithm 392
Review Exercises 397
13
Depth-First Search and Applications 399
13.1 Depth-First Search 399
13.2 The One-Way Street Problem 405
Review Exercises 411
14
Planar Graphs and Colorings 413
14.1 Planar Graphs 413
14.2 Coloring Graphs 421
14.3 Circuit Testing and Facilities Design 430
Review Exercises 438
1 5
The Max Flow-Min Cut Theorem 441
15.1 Flows and Cuts 441
15.2 Constructing Maximal Flows 448
15.3 Applications 454
15.4 Matchings 459
Review Exercises 464
Solutions to Selected Exercises S-i
Glossary G-1
Index I-l
Preface
To the Student from the Authors
Few people ever read a preface, and those who do often just glance at the first few lines. So we
begin by answering the question most frequently asked by the readers of our manuscript: "What
does [BB] mean?" Like most undergraduate texts in mathematics these days, answers to some
of our exercises appear at the back of the book. Those which do are marked [BB] for "Back of
Book." In this book, complete solutions, not simply answers, are given to almost all the exercises
marked [BB]. So, in a sense, there is a free Student Solutions Manual at the end of this text.
We are active mathematicians who have always enjoyed solving problems. It is our hope that
our enthusiasm for mathematics and, in particular, for discrete mathematics is transmitted to our
readers through our writing.
The word "discrete" means separate or distinct. Mathematicians view it as the opposite of
"continuous." Whereas, in calculus, it is continuous functions of a real variable that are important,
such functions are of relatively little interest in discrete mathematics. Instead of the real numbers,
it is the natural numbers 1, 2, 3, ... that play a fundamental role, and it is functions with domain
the natural numbers that are often studied. Perhaps the best way to summarize the subject matter
of this book is to say that discrete mathematics is the study of problems associated with the
natural numbers.
You should never read a mathematics book or notes taken in a mathematics course the way
you read a novel, in an easy chair by the fire. You should read at a desk, with paper and pencil
at hand, verifying statements which are less than clear and inserting question marks in margins
so that you are ready to ask questions at the next available opportunity.
Definitions and terminology are terribly important in mathematics, much more so than many
students think. In our experience, the number one reason why students have difficulty with
"proofs" in mathematics is their failure to understand what the question is asking. This book
contains a glossary of definitions (often including examples) at the end as well as a summary of
notation inside the front and back covers. We urge you to consult these areas of the book regularly.
As an aid to interaction between author and student, we occasionally ask you to "pause a
moment" and think about a specific point we have just raised. Our Pauses are little questions
intended to be solved on the spot, right where they occur, like this.
Where will you find [BB] in this book and what does it mean'?
The answers to Pauses are given at the end of every section just before the exercises. So when
a Pause appears, it is easy to cheat by turning over the page and looking at the answer, but that,
of course, is not the way to learn mathematics!
XD We believe that writing skills are terribly important, so, in this edition, we have highlighted
some exercises where we expect answers to be written in complete sentences and good English.
xi
