Fundamentals of Discrete Mathematical Structures

Transcription

1

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

3 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 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 and Printed by Raj Press, New Delhi

4 To my parents RUKMA and SUJARAM

5

6 Contents Preface... xiii Preface to the First Edition... xv 1. DISCRETE STRUCTURES AND SET THEORY Introduction Defining Discrete Structures The Essence of Set Theory Sets, Members and Subsets Inverse of a Set Ordered Pairs Set Operations Union and Intersection Difference and Symmetric Difference Generalized Intersection and Union De Morgan s Laws Finite and Infinite Sets Uncountably Infinite Sets Russell s Paradox Historical Preview Exercises Multiple Choice Questions INDUCTION, RECURSION AND RECURRENCES Introduction Mathematical Induction Why Mathematical Induction Works? Second Principle of Mathematical Induction Recursion and Recurrences Iterating a Recurrence Recursion Trees Linear Recurrences v

7 vi Contents 2.6 Historical Preview Exercises Multiple Choice Questions COMBINATORICS Introduction Counting Principle Power Set Principle of Inclusion and Exclusion Generalized Principle of Inclusion and Exclusion Pigeonhole Principle Injection, Surjection and Bijection Permutations Combinations Schroeder Bernstein Theorem Historical Preview Exercises Multiple Choice Questions DISCRETE PROBABILITY Introduction Events Discrete Probability Distribution Function Frequency versus Probability Conditional Probability Historical Preview Exercises Multiple Choice Questions MATHEMATICAL LOGIC Introduction Propositional Logic Semantics and Truth Tables Propositional Equivalences Propositional Language Deductions Soundness and Completeness Normal Forms Defining Normal Forms Conversion between Normal Forms... 94

8 Contents vii 5.7 Fuzzy Logic Crisp Sets Fuzzy Sets Historical Preview Exercises Multiple Choice Questions LOGICAL INFERENCING Introduction Inference Rules Historical Preview Exercises Multiple Choice Questions PREDICATE LOGIC Introduction Predicate Formulae Functions Variables and Quantifiers Inference Rules Rule of Universal Instantiation Rule of Universal Generalization Rule of Existential Instantiation Rule of Existential Generalization Scope of Variables Inversion of Quantified Expressions Domain of Discourse The Peano Axioms Iteration Addition Higher Order Predicate Logic Temporal Logic Computer Science Applications Historical Preview Exercises Multiple Choice Questions GRAPH THEORY Introduction Graph Terminology Degrees of Nodes Isomorphic Graphs Dijkstra s Shortest Path Algorithm

9 viii Contents 8.6 Planar Graphs Eulerian Graphs Hamiltonian Graphs Graph Search Breadth-first Search Depth-first Search Travelling Salesman Problem Graph and Map Coloring Vertex Coloring Bipartite Graph Matchings Historical Preview Exercises Multiple Choice Questions RELATIONS Introduction Relations on Sets Computer Representation of Relations Combination Relations Properties of Relations Reflexive and Irreflexive Relations Symmetric, Asymmetric and Antisymmetric Relations Representation and Processing of Relations Using Matrices Closure of Relations Transitive Relation Historical Preview Exercises Multiple Choice Questions TRANSITIVE CLOSURE AND WARSHALL S ALGORITHM Introduction Transitive Closure Complexity of Transitive Closure Algorithm Warshall s Algorithm Historical Preview Exercises Multiple Choice Questions EQUIVALENCE AND PARTIAL ORDERING RELATIONS Introduction Equivalence Relations

10 Contents ix 11.3 Partially Ordered Sets and Relations Topological Sorting Lexicographic Ordering Hasse Diagram Properties of Posets Lattices Chain, Antichains and Order-Isomorphism Complemented Lattices Historical Preview Exercises Multiple Choice Questions TREES Introduction Rooted and Other Trees Representation of Well Formed Formulae Representation of Arithmetic Expressions Representation of Prefix Codes Spanning Trees Kruskal s Algorithm Prim s Algorithm Traversing Binary Trees Binary Search Trees Historical Preview Exercises Multiple Choice Questions ALGEBRAIC SYSTEMS Introduction Algebraic Systems Groupoids Semigroup Isomorphism Homomorphism Congruence Admissible Partitions Relationship between Homomorphism, Congruence and Admissible Partitions Groups Order of a Group Isomorphism of Groups Cyclic Group Group Homomorphism

11 Fundamentals Of Discrete Mathe Structures 30% OFF Publisher : PHI Learning ISBN : Author : K. R. Chowdhary Type the URL : Get this ebook