Foundations of discrete mathematics with algorithms and programming

535 Pages·2019·2.165 MB·English

by  Balakrishnan, Rangaswami, Sridharan, Sriraman

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Foundations of
Discrete Mathematics
with Algorithms and
Programming
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Foundations of
Discrete Mathematics
with Algorithms and
Programming
Sriraman Sridharan
Laboratoire LAMPS
Département de Mathématiques et d’Informatique
Université de Perpignan Via Domitia
Perpignan
FRANCE
R. Balakrishnan
Bharathidasan University
Tiruchirappalli
Tamil Nadu
INDIA
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CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2019 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
Version Date: 20180813
International Standard Book Number-13: 978-0-8153-7848-8 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
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Library of Congress Cataloging-in-Publication Data
Names: Sridharan, Sriraman, author. | Balakrishnan, R. (Rangaswami), author.
Title: Foundations of discrete mathematics with algorithms and programming /
Sriraman Sridharan, R. Balakrishnan.
Description: Boca Raton : Taylor & Francis, a CRC title, part of the Taylor &
Francis imprint, a member of the Taylor & Francis Group, the academic
division of T&F Informa, plc, 2019. | Includes bibliographical references
and index.
Identifiers: LCCN 2018018635| ISBN 9780815378488 (hardback : acid-free paper)
| ISBN 9781351019149 (ebook)
Subjects: LCSH: Computer science--Mathematics.
Classification: LCC QA76.9.M35 S725 2019 | DDC 004.01/51--dc23
LC record available at https://lccn.loc.gov/2018018635
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
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The (cid:12)rst author S. S. affectionately dedicates this book in
memory of his parents:
Geeyar Sriraman
Padmavathi Sundaravaradhan
The second author R. B. dedicates this book in memory of
the Late Professor Jacob K. Goldhaber,
formerlyoftheUniversityofMaryland,USAforallthegoodthings
in life the author learned from him, besides, of course, mathemat-
ics.
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Contents
Preface xi
1 Sets, Relations and Functions 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Equivalence Relations . . . . . . . . . . . . . . . . . 9
1.4 Finite and In(cid:12)nite Sets . . . . . . . . . . . . . . . 11
1.5 Cardinal Numbers of Sets . . . . . . . . . . . . . . 14
1.6 Power Set of a Set . . . . . . . . . . . . . . . . . . 17
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 19
1.8 Partially Ordered Sets . . . . . . . . . . . . . . . . 20
1.9 Lattices . . . . . . . . . . . . . . . . . . . . . . . . 25
1.10 Boolean Algebras . . . . . . . . . . . . . . . . . . . 34
1.10.1 Introduction . . . . . . . . . . . . . . . . . . 34
1.10.2 Examples of Boolean Algebras . . . . . . . . 34
1.11 Atoms in a Lattice . . . . . . . . . . . . . . . . . . 38
1.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Combinatorics 47
2.1 What Is Combinatorics? . . . . . . . . . . . . . . . 48
2.2 Elementary Counting Principles . . . . . . . . . . . 51
2.3 Permutations and Combinations . . . . . . . . . . . 64
2.3.1 Sums and Products . . . . . . . . . . . . . . 79
2.4 Binomial Theorem . . . . . . . . . . . . . . . . . . 83
2.5 Multinomial Coefficients . . . . . . . . . . . . . . . 91
2.6 Stirling Numbers . . . . . . . . . . .{.}. . . . . . . 103
2.7 Stirling Number of the Second Kind n . . . . . . 113
k
vii
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viii CONTENTS
2.8 Bell Numbers . . . . . . . . . . . . . . . . . . . . . 125
2.9 The Principle of Inclusion . . . . . . . . . . . . . . 127
2.9.1 Applications of Inclusion and Exclusion
Principle . . . . . . . . . . . . . . . . . . . . 132
2.9.2 Application of Inclusion and Exclusion
Principle to Elementary Number Theory . . 137
2.9.3 Applications to Permanents . . . . . . . . . 143
2.10 Generating Functions . . . . . . . . . . . . . . . . . 152
2.10.1 Solving Recurrence Relations Using
Generating Function Techniques . . . . . . . 157
2.10.2 Catalan Numbers . . . . . . . . . . . . . . . 160
2.11 Generating Subsets . . . . . . . . . . . . . . . . . . 166
2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 175
3 Basics of Number Theory 183
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 184
3.2 Divisibility . . . . . . . . . . . . . . . . . . . . . . . 184
3.3 gcd and lcm of Two Integers . . . . . . . . . . . . . 186
3.4 Primes . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 193
3.6 Congruences . . . . . . . . . . . . . . . . . . . . . . 194
3.7 Complete System of Residues . . . . . . . . . . . . 197
3.8 Linear Congruences . . . . . . . . . . . . . . . . . . 202
3.9 Lattice Points Visible from the Origin . . . . . . . . 206
3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 209
3.11 Some Arithmetical Functions . . . . . . . . . . . . 210
3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 218
3.13 The Big O Notation . . . . . . . . . . . . . . . . . 218
4 Introduction to Graph Theory 225
4.1 The Idea of a Graph . . . . . . . . . . . . . . . . . 225
4.2 An Informal and Intuitive . . . . . . . . . . . . . . 234
4.3 Multigraph or Undirected Graph . . . . . . . . . . 239
4.4 Some Special Graphs . . . . . . . . . . . . . . . . . 244
4.5 Graphs and Subgraphs . . . . . . . . . . . . . . . . 264
4.6 Walks, Paths, Cycles . . . . . . . . . . . . . . . . . 271
4.7 Connectedness . . . . . . . . . . . . . . . . . . . . . 275
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CONTENTS ix
4.8 Graphs and Puzzles . . . . . . . . . . . . . . . . . . 283
4.8.1 An Application . . . . . . . . . . . . . . . . 288
4.8.2 Two Friendship Theorems . . . . . . . . . . 289
4.8.3 Pandava Princes Problem and 3 Houses, 3
Utilities Problem . . . . . . . . . . . . . . . 292
4.9 Ramsey Numbers . . . . . . . . . . . . . . . . . . . 293
4.10 Graph Algebra . . . . . . . . . . . . . . . . . . . . 308
4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 314
5 Introduction to Algorithms 325
5.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . 326
5.2 Complexity of Algorithms . . . . . . . . . . . . . . 330
5.3 An Overview of a Classical Computer . . . . . . . . 353
5.4 Introduction to Programming . . . . . . . . . . . . 360
5.4.1 Parameter Passing . . . . . . . . . . . . . . 373
5.4.2 Recursion . . . . . . . . . . . . . . . . . . . 381
5.5 Introduction to Data Structures . . . . . . . . . . . 387
5.5.1 Access Restricted Lists . . . . . . . . . . . . 406
5.6 Examples of Algorithms . . . . . . . . . . . . . . . 416
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 439
6 Introduction to Logic 447
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 447
6.2 Algebra of Propositions . . . . . . . . . . . . . . . . 448
6.3 Proofs in Mathematics . . . . . . . . . . . . . . . . 454
6.3.1 Universal and Existential Quanti(cid:12)ers . . . . 457
6.4 Probability . . . . . . . . . . . . . . . . . . . . . . 458
Appendices Answers to Even-Numbered Exercises 477
A Answers to Chapter 1 479
B Answers to Chapter 2 483
C Answers to Chapter 3 487
D Answers to Chapter 4 491
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