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Discrete
Mathematics
Applications
and Its
Kenneth H. Rosen
Eighth Edition
Discrete
Mathematics
and Its
Applications
Eighth Edition
Kenneth H. Rosen
formerly AT&T Laboratories
DISCRETE MATHEMATICS AND ITS APPLICATIONS, EIGHTH EDITION
Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright
c
2019 by McGraw-Hill Education. All rights reserved. Printed
in the United States of America. Previous editions
c
2012, 2007, and 2003. No part of this publication may be reproduced or distributed in any form or
by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, inany
network or other electronic storage or transmission, or broadcast for distance learning.
Some ancillaries, including electronic and print components, may not be available to customers outside the United States.
This book is printed on acid-free paper.
123456789LWI21201918
ISBN 978-1-259-67651-2
MHID 1-259-67651-X
Product Developer: Nora Devlin
Marketing Manager: Alison Frederick
Content Project Manager: Peggy Selle
Buyer: Sandy Ludovissy
Design: Egzon Shaqiri
Content Licensing Specialist: Lorraine Buczek
Cover Image:
c
Karl Dehnam/Alamy Stock Photo
Compositor: Aptara, Inc.
All credits appearing on 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
Names: Rosen, Kenneth H., author.
Title: Discrete mathematics and its applications / Kenneth H. Rosen, Monmouth
University (and formerly AT&T Laboratories).
Description: Eighth edition. |New York, NY : McGraw-Hill, [2019] |Includes
bibliographical references and index.
Identifiers: LCCN 2018008740|ISBN 9781259676512 (alk. paper) |
ISBN 125967651X (alk. paper)
Subjects: LCSH: Mathematics. |Computer science–Mathematics.
Classification: LCC QA39.3 .R67 2019 |DDC 511–dc23 LC record available at
https://lccn.loc.gov/2018008740
The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the
authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites.
mheducation.com/highered
Contents
About the Author vi
Preface vii
Online Resources xvi
To the Student xix
1 The Foundations: Logic and Proofs ....................................1
1.1 Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Applications of Propositional Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
1.3 Propositional Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 Predicates and Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Nested Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.6 Rules of Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
1.7 Introduction to Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
1.8 Proof Methods and Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
End-of-Chapter Material .....................................................115
2 Basic Structures: Sets, Functions, Sequences, Sums,
and Matrices .......................................................121
2.1 Sets........................................................................121
2.2 Set Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.4 Sequences and Summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
2.5 Cardinality of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
2.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
End-of-Chapter Material .....................................................195
3 Algorithms .........................................................201
3.1 Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.2 The Growth of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
3.3 Complexity of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
End-of-Chapter Material .....................................................244
4 Number Theory and Cryptography ..................................251
4.1 Divisibility and Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.2 Integer Representations and Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.3 Primes and Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
4.4 Solving Congruences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .290
4.5 Applications of Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
4.6 Cryptography...............................................................310
End-of-Chapter Material .....................................................324
iii
iv Contents
5 Induction and Recursion ............................................331
5.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
5.2 Strong Induction and Well-Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
5.3 Recursive Definitions and Structural Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
5.4 Recursive Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
5.5 Program Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
End-of-Chapter Material .....................................................398
6 Counting ...........................................................405
6.1 The Basics of Counting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .405
6.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
6.3 Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
6.4 Binomial Coefficients and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
6.5 Generalized Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
6.6 Generating Permutations and Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
End-of-Chapter Material .....................................................461
7 Discrete Probability .................................................469
7.1 An Introduction to Discrete Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
7.2 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
7.3 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
7.4 Expected Value and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
End-of-Chapter Material .....................................................520
8 Advanced Counting Techniques .....................................527
8.1 Applications of Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
8.2 Solving Linear Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
8.3 Divide-and-Conquer Algorithms and Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . 553
8.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
8.5 Inclusion–Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
8.6 Applications of Inclusion–Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
End-of-Chapter Material .....................................................592
9Relations...........................................................599
9.1 Relations and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
9.2 n-ary Relations and Their Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .611
9.3 Representing Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621
9.4 Closures of Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .628
9.5 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
9.6 Partial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
End-of-Chapter Material .....................................................665
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