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Fundamental approach to discrete mathematics acharjya DP (2005)

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Cấu trúc

  • Cover

  • Preface

  • Contents

  • Chapter 1. Mathematical Logic

    • 1.0 Introduction

    • 1.1 Statement (Proposition)

    • 1.2 Logical Connectives

    • 1.3 Conditional

    • 1.4 Bi-Conditional

    • 1.5 Converse

    • 1.6 Inverse

    • 1.7 Contra Positive

    • 1.8 Exclusive OR

    • 1.9 NAND

    • 1.10 NOR

    • 1.11 Tautology

    • 1.12 Contradiction

    • 1.13 Satisfiable

    • 1.14 Duality Law

    • 1.15 Algebra of Propositions

    • 1.16 Mathematical Induction

    • Solved Examples

    • Exercises

  • Chapter 2. Set Theory

    • 2.0 Introduction

    • 2.1 Sets

    • 2.2 Types of Sets

    • 2.3 Cardinality of a Set

    • 2.4 Subset and Superset

    • 2.5 Comparability of Sets

    • 2.6 Power Set

    • 2.7 Operations on Sets

    • 2.8 Disjoint Sets

    • 2.9 Application of Set Theory

    • 2.10 Product of Sets

    • 2.11 Fundamental Products

    • Solved Examples

    • Exercises

  • Chapter 3. Binary Relation

    • 3.0 Introduction

    • 3.1 Binary Relation

    • 3.2 Inverse Relation

    • 3.3 Graph of Relation

    • 3.4 Kind of Relation

    • 3.5 Arrow Diagram

    • 3.6 Void Relation

    • 3.7 Identity Relation

    • 3.8 Universal Relation

    • 3.9 Relation Matrix (Matrix of the Relation)

    • 3.10 Composition of Relations

    • 3.11 Types of Relations

    • 3.12 Types of Relations and Relation Matrix

    • 3.13 Equivalence Relation

    • 3.14 Partial Order Relation

    • 3.15 Total Order Relation

    • 3.16 Closures of Relations

    • 3.17 Equivalence Classes

    • 3.18 Partitions

    • Solved Examples

    • Exercises

  • Chapter 4. Function

    • 4.0 Introduction

    • 4.1 Function

    • 4.2 Equality of Functions

    • 4.3 Types of Function

    • 4.4 Graph of Function

    • 4.5 Composition of Functions

    • 4.6 Inverse Function

    • 4.7 Some Important Functions

    • 4.8 Hash Function

    • Solved Examples

    • Exercises

  • Chapter 5. Group Theory

    • 5.0 Introduction

    • 5.1 Binary Operation on a Set

    • 5.2 Algebraic Structure

    • 5.3 Group

    • 5.4 Subgroup

    • 5.5 Cyclic Group

    • 5.6 Cosets

    • 5.7 Homomorphism

    • Solved Examples

    • Exercises

  • Chapter 6. Codes and Group Codes

    • 6.0 Introduction

    • 6.1 Terminologies

    • 6.2 Error Correction

    • 6.3 Group Codes

    • 6.4 Weight of Code Word

    • 6.5 Distance Between the Code Words

    • 6.6 Error Correction for Block Code

    • 6.7 Cosets

    • Solved Examples

    • Exercises

  • Chapter 7. Ring Theory

    • 7.0 Introduction

    • 7.1 Ring

    • 7.2 Special Types of Ring

    • 7.3 Ring without Zero Divisor

    • 7.4 Integral Domain

    • 7.5 Division Ring

    • 7.6 Field

    • 7.7 The Pigeonhole Principle

    • 7.8 Characteristics of a Ring

    • 7.9 Sub Ring

    • 7.10 Homomorphism

    • 7.11 Kernel of Homomorphism of Ring

    • 7.12 Isomorphism

    • Solved Examples

    • Exercises

  • Chapter 8. Boolean Algebra

    • 8.0 Introduction

    • 8.1 Gates

    • 8.2 More Logic Gates

    • 8.3 Combinatorial Circuit

    • 8.4 Boolean Expression

    • 8.5 Equivalent Combinatorial Circuits

    • 8.6 Boolean Algebra

    • 8.7 Dual of a Statement

    • 8.8 Boolean Function

    • 8.9 Various Normal Forms

    • Solved Examples

    • Exercises

  • Chapter 9. Introduction to Lattices

    • 9.0 Introduction

    • 9.1 Lattices

    • 9.2 Hasse Diagram

    • 9.3 Principle of Duality

    • 9.4 Distributive Lattice

    • 9.5 Bounded Lattice

    • 9.6 Complemented Lattice

    • 9.7 Some Special Lattices

    • Solved Examples

    • Exercises

  • Chapter 10. Graph Theory

    • 10.0 Introduction

    • 10.1 Graph

    • 10.2 Kinds of Graph

    • 10.3 Digraph

    • 10.4 Weighted Graph

    • 10.5 Degree of a Vertex

    • 10.6 Path

    • 10.7 Complete Graph

    • 10.8 Regular Graph

    • 10.9 Cycle

    • 10.10 Pendant Vertex

    • 10.11 Acyclic Graph

    • 10.12 Matrix Representation of Graphs

    • 10.13 Connected Graph

    • 10.14 Graph Isomorphism

    • 10.15 Bipartite Graph

    • 10.16 Subgraph

    • 10.17 Walks

    • 10.18 Operations on Graphs

    • 10.19 Fusion of Graphs

    • Solved Examples

    • Exercises

  • Chapter 11. Tree

    • 11.0 Introduction

    • 11.1 Tree

    • 11.2 Fundamental Terminologies

    • 11.3 Binary Tree

    • 11.4 Bridge

    • 11.5 Distance

    • 11.6 Eccentricity

    • 11.7 Radius

    • 11.8 Diameter

    • 11.9 Central Point and Centre

    • 11.10 Spanning Tree

    • 11.11 Searching Algorithms

    • 11.12 Shortest Path Algorithms

    • 11.13 Cut Vertices

    • 11.14 Euler Graph

    • 11.15 Hamiltonian Path

    • 11.16 Closure of a Graph

    • 11.17 Travelling Salesman Problem

    • Solved Examples

    • Exercises

Nội dung

THIS PAGE IS BLANK NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS PUBLISHING FOR ONE WORLD New Delhi · Bangalore · Chennai · Cochin · Guwahati · Hyderabad Jalandhar · Kolkata · Lucknow · Mumbai · Ranchi Visit us at www.newagepublishers.com Copyright © 2005, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher All inquiries should be emailed to rights@newagepublishers.com ISBN (10) : 81-224-2304-3 ISBN (13) : 978-81-224-2304-4 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com THIS PAGE IS BLANK # Fundamental Approach to Discrete Mathematics Example algorithm For the following travelling salesman problem, carry out the closest insertion Solution : Given that the complete weighted graph G as Choose the vertex v1 Choose the vertex v2, which is closest to v1 So, w2 = v1 v2 v1 Choose the vertex v 3, which is close to w2 So, w3 = v v2 v3 v1 Choose the vertex v4, which is close to w3 Hence, we have the following cases w4 = v1 v2 v3 v4 v1 or = v1 v v4 v3 v1 or = v1 v4 v2 v3 v Now length of v1 v2 v3 v4 v1 = 10 + 40 + 30 + 20 = 100 Length of v1 v2 v4 v3 v1 = 10 + 45 + 30 + 15 = 100 Length of v1 v4 v2 v3 v1 = 20 + 45 + 40 + 15 = 120 Therefore, w4 = v1 v2 v3 v4 v1 is minimum Choose the vertex v5, which is close to w4 Hence, we have the following cases The length of following cycles is given as below v1 v2 v3 v4 v5 v1 = 10 + 40 + 30 + 55 + 25 = 160 v1 v2 v3 v5 v4 v1 = 10 + 40 + 35 + 55 + 20 = 160 v1 v2 v5 v3 v4 v1 = 10 + 50 + 35 + 30 + 20 = 145 v1 v5 v2 v3 v4 v1 = 25 + 50 + 40 + 30 + 20 = 165 As all the vertices are included in the cycle, so the process terminates Hence , the shortest Hamiltonian cycle is given as v1 v2 v5 v v4 v1 Tree # Example For the travelling salesman problem given in example 4, carry out the two optimal algorithm Solution : For the complete weighted graph G given above, the number of vertices (n) = According to the two optimal algorithm we have the following steps Let C = v1, v2, v3, v4, v5, v1 be a Hamiltonian cycle Therefore, we get w = w(v1v2) + w(v2v3) + w(v3v4) + w(v4v5) + w(v5v1) = 10 + 40 + 30 + 55 + 25 = 160 Set i = Set j = i + = Set Cij = C13 = v1 v3 v2 v4 v5 v1 w13 = w – w(v1v2) – w(v3v4) + w(v1v3) + w(v2v4) = 160 – 10 – 30 + 15 + 45 = 180 As w13

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