Fundamentals of Discrete Mathematical Structures

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Fundamentals of Discrete Mathematical Structures THIRD EDITION K.R. Chowdhary Campus Director JIET School of Engineering and Technology for Girls Jodhpur Delhi-110092 2015

FUNDAMENTALS OF DISCRETE MATHEMATICAL STRUCTURES, Third Edition K.R. Chowdhary 2015 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-5074-8 The export rights of this book are vested solely with the publisher. Fourth Printing (Third Edition) January, 2015 Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Raj Press, New Delhi-110012.

To my parents RUKMA and SUJARAM

Contents Preface... xiii Preface to the First Edition... xv 1. DISCRETE STRUCTURES AND SET THEORY... 1 25 1.1 Introduction... 1 1.2 Defining Discrete Structures... 2 1.3 The Essence of Set Theory... 2 1.4 Sets, Members and Subsets... 3 1.4.1 Inverse of a Set... 5 1.4.2 Ordered Pairs... 6 1.5 Set Operations... 6 1.5.1 Union and Intersection... 6 1.5.2 Difference and Symmetric Difference... 9 1.6 Generalized Intersection and Union... 11 1.7 De Morgan s Laws... 11 1.8 Finite and Infinite Sets... 14 1.9 Uncountably Infinite Sets... 16 1.10 Russell s Paradox... 18 1.11 Historical Preview... 19 Exercises... 21 Multiple Choice Questions... 24 2. INDUCTION, RECURSION AND RECURRENCES... 26 41 2.1 Introduction... 26 2.2 Mathematical Induction... 26 2.3 Why Mathematical Induction Works?... 28 2.4 Second Principle of Mathematical Induction... 32 2.5 Recursion and Recurrences... 33 2.5.1 Iterating a Recurrence... 36 2.5.2 Recursion Trees... 36 2.5.3 Linear Recurrences... 37 v

vi Contents 2.6 Historical Preview... 38 Exercises... 39 Multiple Choice Questions... 41 3. COMBINATORICS... 42 68 3.1 Introduction... 42 3.2 Counting Principle... 43 3.3 Power Set... 46 3.4 Principle of Inclusion and Exclusion... 47 3.5 Generalized Principle of Inclusion and Exclusion... 49 3.6 Pigeonhole Principle... 52 3.7 Injection, Surjection and Bijection... 54 3.8 Permutations... 55 3.9 Combinations... 58 3.10 Schroeder Bernstein Theorem... 60 3.11 Historical Preview... 62 Exercises... 63 Multiple Choice Questions... 67 4. DISCRETE PROBABILITY...69 83 4.1 Introduction... 69 4.2 Events... 69 4.3 Discrete Probability... 72 4.3.1 Distribution Function... 72 4.4 Frequency versus Probability... 76 4.5 Conditional Probability... 77 4.6 Historical Preview... 80 Exercises... 81 Multiple Choice Questions... 82 5. MATHEMATICAL LOGIC...84 106 5.1 Introduction... 84 5.2 Propositional Logic... 85 5.3 Semantics and Truth Tables... 86 5.4 Propositional Equivalences... 88 5.5 Propositional Language... 89 5.5.1 Deductions... 90 5.5.2 Soundness and Completeness... 91 5.6 Normal Forms... 91 5.6.1 Defining Normal Forms... 92 5.6.2 Conversion between Normal Forms... 94

Contents vii 5.7 Fuzzy Logic... 97 5.7.1 Crisp Sets... 99 5.7.2 Fuzzy Sets... 99 5.8 Historical Preview... 100 Exercises... 103 Multiple Choice Questions... 105 6. LOGICAL INFERENCING...107 116 6.1 Introduction... 107 6.2 Inference Rules... 108 6.3 Historical Preview... 112 Exercises... 114 Multiple Choice Questions... 116 7. PREDICATE LOGIC...117 136 7.1 Introduction... 117 7.2 Predicate Formulae... 118 7.3 Functions... 119 7.4 Variables and Quantifiers... 120 7.5 Inference Rules... 121 7.5.1 Rule of Universal Instantiation... 121 7.5.2 Rule of Universal Generalization... 121 7.5.3 Rule of Existential Instantiation... 122 7.5.4 Rule of Existential Generalization... 122 7.6 Scope of Variables... 123 7.7 Inversion of Quantified Expressions... 124 7.8 Domain of Discourse... 124 7.9 The Peano Axioms... 126 7.9.1 Iteration... 128 7.9.2 Addition... 129 7.10 Higher Order Predicate Logic... 130 7.11 Temporal Logic... 131 7.11.1 Computer Science Applications... 132 7.12 Historical Preview... 133 Exercises... 134 Multiple Choice Questions... 135 8. GRAPH THEORY...137 164 8.1 Introduction... 137 8.2 Graph Terminology... 138 8.3 Degrees of Nodes... 140 8.4 Isomorphic Graphs... 142 8.5 Dijkstra s Shortest Path Algorithm... 143

viii Contents 8.6 Planar Graphs... 148 8.7 Eulerian Graphs... 149 8.8 Hamiltonian Graphs... 151 8.9 Graph Search... 152 8.9.1 Breadth-first Search... 152 8.9.2 Depth-first Search... 154 8.10 Travelling Salesman Problem... 155 8.11 Graph and Map Coloring... 156 8.11.1 Vertex Coloring... 157 8.12 Bipartite Graph... 158 8.13 Matchings... 158 8.14 Historical Preview... 159 Exercises... 161 Multiple Choice Questions... 163 9. RELATIONS...165 189 9.1 Introduction... 165 9.2 Relations on Sets... 165 9.3 Computer Representation of Relations... 168 9.4 Combination Relations... 170 9.5 Properties of Relations... 171 9.5.1 Reflexive and Irreflexive Relations... 171 9.5.2 Symmetric, Asymmetric and Antisymmetric Relations... 172 9.6 Representation and Processing of Relations Using Matrices... 174 9.7 Closure of Relations... 177 9.8 Transitive Relation... 179 9.9 Historical Preview... 181 Exercises... 183 Multiple Choice Questions... 187 10. TRANSITIVE CLOSURE AND WARSHALL S ALGORITHM...190 201 10.1 Introduction... 190 10.2 Transitive Closure... 191 10.3 Complexity of Transitive Closure Algorithm... 195 10.4 Warshall s Algorithm... 196 10.5 Historical Preview... 199 Exercises... 200 Multiple Choice Questions... 201 11. EQUIVALENCE AND PARTIAL ORDERING RELATIONS... 202 222 11.1 Introduction... 202 11.2 Equivalence Relations... 202

Contents ix 11.3 Partially Ordered Sets and Relations... 205 11.3.1 Topological Sorting... 206 11.3.2 Lexicographic Ordering... 207 11.4 Hasse Diagram... 209 11.5 Properties of Posets... 211 11.6 Lattices... 212 11.7 Chain, Antichains and Order-Isomorphism... 214 11.8 Complemented Lattices... 216 11.9 Historical Preview... 217 Exercises... 218 Multiple Choice Questions... 221 12. TREES...223 245 12.1 Introduction... 223 12.2 Rooted and Other Trees... 226 12.3 Representation of Well Formed Formulae... 227 12.4 Representation of Arithmetic Expressions... 229 12.5 Representation of Prefix Codes... 230 12.6 Spanning Trees... 232 12.6.1 Kruskal s Algorithm... 234 12.6.2 Prim s Algorithm... 235 12.7 Traversing Binary Trees... 237 12.8 Binary Search Trees... 238 12.9 Historical Preview... 240 Exercises... 241 Multiple Choice Questions... 244 13. ALGEBRAIC SYSTEMS...246 277 13.1 Introduction... 246 13.2 Algebraic Systems... 247 13.3 Groupoids... 247 13.3.1 Semigroup... 248 13.3.2 Isomorphism... 248 13.3.3 Homomorphism... 249 13.4 Congruence... 252 13.5 Admissible Partitions... 252 13.6 Relationship between Homomorphism, Congruence and Admissible Partitions... 254 13.7 Groups... 256 13.7.1 Order of a Group... 259 13.7.2 Isomorphism of Groups... 262 13.7.3 Cyclic Group... 262 13.7.4 Group Homomorphism... 264

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