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Discrete Mathematics and Its Applications Seventh Edition.pdf
Cover Page
Title Page
Copyright Page
Contents
About the Author
Preface
The Companion Website
To the Student
1 The Foundations: Logic and Proofs
1.1 Propositional Logic
1.2 Applications of Propositional Logic
1.3 Propositional Equivalences
1.4 Predicates and Quantifiers
1.5 Nested Quantifiers
1.6 Rules of Inference
1.7 Introduction to Proofs
1.8 Proof Methods and Strategy
End-of-Chapter Material
2 Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
2.1 Sets
2.2 Set Operations
2.3 Functions
2.4 Sequences and Summations
2.5 Cardinality of Sets
2.6 Matrices
End-of-Chapter Material
3 Algorithms
3.1 Algorithms
3.2 The Growth of Functions
3.3 Complexity of Algorithms
End-of-Chapter Material
4 Number Theory and Cryptography
4.1 Divisibility and Modular Arithmetic
4.2 Integer Representations and Algorithms
4.3 Primes and Greatest Common Divisors
4.4 Solving Congruences
4.5 Applications of Congruences
4.6 Cryptography
End-of-Chapter Material
5 Induction and Recursion
5.1 Mathematical Induction
5.2 Strong Induction andWell-Ordering
5.3 Recursive Definitions and Structural Induction
5.4 Recursive Algorithms
5.5 Program Correctness
End-of-Chapter Material
6 Counting
6.1 The Basics of Counting
6.2 The Pigeonhole Principle
6.3 Permutations and Combinations
6.4 Binomial Coefficients and Identities
6.5 Generalized Permutations and Combinations
6.6 Generating Permutations and Combinations
End-of-Chapter Material
7 Discrete Probability
7.1 An Introduction to Discrete Probability
7.2 Probability Theory
7.3 Bayes’ Theorem
7.4 Expected Value and Variance
End-of-Chapter Material
8 Advanced Counting Techniques
8.1 Applications of Recurrence Relations
8.2 Solving Linear Recurrence Relations
8.3 Divide-and-Conquer Algorithms and Recurrence Relations
8.4 Generating Functions
8.5 Inclusion–Exclusion
8.6 Applications of Inclusion–Exclusion
End-of-Chapter Material
9 Relations
9.1 Relations and Their Properties
9.2 n-ary Relations and Their Applications
9.3 Representing Relations
9.4 Closures of Relations
9.5 Equivalence Relations
9.6 Partial Orderings
End-of-Chapter Material
10 Graphs
10.1 Graphs and Graph Models
10.2 Graph Terminology and Special Types of Graphs
10.3 Representing Graphs and Graph Isomorphism
10.4 Connectivity
10.5 Euler and Hamilton Paths
10.6 Shortest-Path Problems
10.7 Planar Graphs
10.8 Graph Coloring
End-of-Chapter Material
11 Trees
11.1 Introduction to Trees
11.2 Applications of Trees
11.3 Tree Traversal
11.4 Spanning Trees
11.5 Minimum Spanning Trees
End-of-Chapter Material
12 Boolean Algebra
12.1 Boolean Functions
12.2 Representing Boolean Functions
12.3 Logic Gates
12.4 Minimization of Circuits
End-of-Chapter Material
13 Modeling Computation
13.1 Languages and Grammars
13.2 Finite-State Machines with Output
13.3 Finite-State Machines with No Output
13.4 Language Recognition
13.5 Turing Machines
End-of-Chapter Material
Appendixes
1 Axioms for the Real Numbers and the Positive Integers
2 Exponential and Logarithmic Functions
3 Pseudocode
Suggested Readings
Answers to Odd-Numbered Exercises
Photo Credits
Index of Biographies
Index
Kenneth H. Rosen
Discrete
Mathematics
and Its
Applications
SEVENTH EDITION
Discrete
Mathematics
and Its
Applications
Seventh Edition
Kenneth H. Rosen
Monmouth University
(and formerly AT&T Laboratories)
DISCRETE MATHEMATICS AND ITS APPLICATIONS, SEVENTH EDITION
Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the
Americas, New York, NY 10020. Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
Previous editions © 2007, 2003, and 1999. No part of this publication may be reproduced or distributed
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Some ancillaries, including electronic and print components, may not be available to customers outside the
United States.
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1 2 3 4 5 6 7 8 9 0 DOW/DOW 1 0 9 8 7 6 5 4 3 2 1
ISBN 978-0-07-338309-5
MHID 0-07-338309-0
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All credits appearing on this page or at the end of the book are considered to be an extension of the copyright page.
Library of Congress Cataloging-in-Publication Data
Rosen, Kenneth H.
Discrete mathematics and its applications / Kenneth H. Rosen. — 7th ed.
p. cm.
Includes index.
ISBN 0–07–338309–0
1. Mathematics. 2. Computer science—Mathematics. I. Title.
QA39.3.R67 2012
511–dc22
www.mhhe.com
2011011060
Contents
About the Author vi
Preface vii
The Companion Website xvi
To the Student xvii
1
The Foundations: Logic and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1
Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Applications of Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Propositional Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3
1.4
Predicates and Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5 Nested Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.6 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Introduction to Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1.7
1.8
Proof Methods and Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2
Basic Structures: Sets, Functions, Sequences, Sums, and Matrices . 115
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.1
Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.2
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.3
2.4
Sequences and Summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
2.5 Cardinality of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
2.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.2
The Growth of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
3.3 Complexity of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
4 Number Theory and Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4.1 Divisibility and Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Integer Representations and Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
4.2
Primes and Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
4.3
4.4
Solving Congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274
4.5 Applications of Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
4.6 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
iii
iv Contents
5
Induction and Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.2
Strong Induction and Well-Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
5.3 Recursive Definitions and Structural Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
5.4 Recursive Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Program Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
5.5
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
6
Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
The Basics of Counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385
6.1
The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.2
6.3
Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
6.4 Binomial Coefficients and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
6.5 Generalized Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
6.6 Generating Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
7 Discrete Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
7.1 An Introduction to Discrete Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
7.2
Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
7.3 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
Expected Value and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
7.4
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
8 Advanced Counting Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
8.1 Applications of Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
8.2
Solving Linear Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
8.3 Divide-and-Conquer Algorithms and Recurrence Relations. . . . . . . . . . . . . . . . . . . . . . .527
8.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
8.5
Inclusion–Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
8.6 Applications of Inclusion–Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
9
Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
9.1 Relations and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
9.2
n-ary Relations and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
9.3 Representing Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
9.4 Closures of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
9.5
Partial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
9.6
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
Contents v
10 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
10.1 Graphs and Graph Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
10.2 Graph Terminology and Special Types of Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .651
10.3 Representing Graphs and Graph Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
10.4 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
10.5 Euler and Hamilton Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
10.6 Shortest-Path Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
10.7 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
10.8 Graph Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735
11 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
11.1 Introduction to Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
11.2 Applications of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
11.3 Tree Traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772
11.4 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
11.5 Minimum Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803
12 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811
12.1 Boolean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811
12.2 Representing Boolean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819
12.3 Logic Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822
12.4 Minimization of Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843
13 Modeling Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847
13.1 Languages and Grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847
13.2 Finite-State Machines with Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .858
13.3 Finite-State Machines with No Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865
13.4 Language Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878
13.5 Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888
End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899
Appendixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1
1
2
3
Axioms for the Real Numbers and the Positive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Suggested Readings B-1
Answers to Odd-Numbered Exercises S-1
Photo Credits C-1
Index of Biographies I-1
Index I-2
About the Author
Kenneth H. Rosen has had a long career as a Distinguished Member of the Technical Staff
at AT&T Laboratories in Monmouth County, New Jersey. He currently holds the position
of Visiting Research Professor at Monmouth University, where he teaches graduate courses in
computer science.
Dr. Rosen received his B.S. in Mathematics from the University of Michigan, Ann Arbor
(1972), and his Ph.D. in Mathematics from M.I.T. (1976), where he wrote his thesis in the area
of number theory under the direction of Harold Stark. Before joining Bell Laboratories in 1982,
he held positions at the University of Colorado, Boulder; The Ohio State University, Columbus;
and the University of Maine, Orono, where he was an associate professor of mathematics.
While working at AT&T Labs, he taught at Monmouth University, teaching courses in discrete
mathematics, coding theory, and data security. He currently teaches courses in algorithm design
and in computer security and cryptography.
Dr. Rosen has published numerous articles in professional journals in number theory and
in mathematical modeling. He is the author of the widely used Elementary Number Theory and
Its Applications, published by Pearson, currently in its sixth edition, which has been translated
into Chinese. He is also the author of Discrete Mathematics and Its Applications, published by
McGraw-Hill, currently in its seventh edition. Discrete Mathematics and Its Applications has
sold more than 350,000 copies in North America during its lifetime, and hundreds of thousands
of copies throughout the rest of the world. This book has also been translated into Spanish,
French, Greek, Chinese, Vietnamese, and Korean. He is also co-author of UNIX: The Complete
Reference; UNIX System V Release 4: An Introduction; and Best UNIX Tips Ever, all published by
Osborne McGraw-Hill. These books have sold more than 150,000 copies, with translations into
Chinese, German, Spanish, and Italian. Dr. Rosen is also the editor of the Handbook of Discrete
and Combinatorial Mathematics, published by CRC Press, and he is the advisory editor of the
CRC series of books in discrete mathematics, consisting of more than 55 volumes on different
aspects of discrete mathematics, most of which are introduced in this book. Dr. Rosen serves as an
Associate Editor for the journal Discrete Mathematics, where he works with submitted papers in
several areas of discrete mathematics, including graph theory, enumeration, and number theory.
He is also interested in integrating mathematical software into the educational and professional
environments, and worked on several projects with Waterloo Maple Inc.’s MapleTM software
in both these areas. Dr. Rosen has also worked with several publishing companies on their
homework delivery platforms.
At Bell Laboratories and AT&T Laboratories, Dr. Rosen worked on a wide range of projects,
including operations research studies, product line planning for computers and data communi-
cations equipment, and technology assessment. He helped plan AT&T’s products and services in
the area of multimedia, including video communications, speech recognition, speech synthesis,
and image networking. He evaluated new technology for use by AT&T and did standards work
in the area of image networking. He also invented many new services, and holds more than 55
patents. One of his more interesting projects involved helping evaluate technology for the AT&T
attraction that was part of EPCOT Center.
vi
Preface
In writing this book, I was guided by my long-standing experience and interest in teaching
discrete mathematics. For the student, my purpose was to present material in a precise,
readable manner, with the concepts and techniques of discrete mathematics clearly presented
and demonstrated. My goal was to show the relevance and practicality of discrete mathematics
to students, who are often skeptical. I wanted to give students studying computer science all of
the mathematical foundations they need for their future studies. I wanted to give mathematics
students an understanding of important mathematical concepts together with a sense of why
these concepts are important for applications. And most importantly, I wanted to accomplish
these goals without watering down the material.
For the instructor, my purpose was to design a flexible, comprehensive teaching tool using
proven pedagogical techniques in mathematics. I wanted to provide instructors with a package
of materials that they could use to teach discrete mathematics effectively and efficiently in the
most appropriate manner for their particular set of students. I hope that I have achieved these
goals.
I have been extremely gratified by the tremendous success of this text. The many improve-
ments in the seventh edition have been made possible by the feedback and suggestions of a large
number of instructors and students at many of the more than 600 North American schools, and
at any many universities in parts of the world, where this book has been successfully used.
This text is designed for a one- or two-term introductory discrete mathematics course taken
by students in a wide variety of majors, including mathematics, computer science, and engineer-
ing. College algebra is the only explicit prerequisite, although a certain degree of mathematical
maturity is needed to study discrete mathematics in a meaningful way. This book has been de-
signed to meet the needs of almost all types of introductory discrete mathematics courses. It is
highly flexible and extremely comprehensive. The book is designed not only to be a successful
textbook, but also to serve as valuable resource students can consult throughout their studies
and professional life.
Goals of a Discrete Mathematics Course
A discrete mathematics course has more than one purpose. Students should learn a particular
set of mathematical facts and how to apply them; more importantly, such a course should teach
students how to think logically and mathematically. To achieve these goals, this text stresses
mathematical reasoning and the different ways problems are solved. Five important themes
are interwoven in this text: mathematical reasoning, combinatorial analysis, discrete structures,
algorithmic thinking, and applications and modeling. A successful discrete mathematics course
should carefully blend and balance all five themes.
1. Mathematical Reasoning: Students must understand mathematical reasoning in order to
read, comprehend, and construct mathematical arguments. This text starts with a discussion
of mathematical logic, which serves as the foundation for the subsequent discussions of
methods of proof. Both the science and the art of constructing proofs are addressed. The
technique of mathematical induction is stressed through many different types of examples
of such proofs and a careful explanation of why mathematical induction is a valid proof
technique.
vii
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