Math Companion for Computer Science

388 Pages·1982·172.407 MB·English

by  Zamir Bavel

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Math Companion for Computer Science

Math Companion for Computer Science
Zamir Bavel
RESTON PUBLISHING COMPANY, INC.
A Prentice-Hall Company
Reston, Virginia
Library of Congress Cataloging in Publication Data
Bavel, Zamir, 1929-
Math companion for computer science.
Bibliography: p.
Includes index.
1. Mathematics—1961- . 2. Electronic data
processing —Mathematics. I. Title.
QA39.2.B38 510 81-22650
ISBN 0-8359-4300-3 AACR2
ISBN 0-8359-4299-6 (pbk.)
Interior design and editing/production supervision
by Ginger Sasser
© 1982 by
Reston Publishing Company, Inc.
A Prentice-Hall Company
Reston, Virginia
All rights reserved. No part of this book may
be reproduced in any way, or by any means, without
permission in writing from the publisher.
10 9 8 7 6 5 4 3 2
Printed in the United States of America
Dedicated to
my father
and to the memory of my mother

Contents
List 0f SYMDOIS........ooiiiirieiiieineeeeece et xv
Preface....ouoieieeiieeee ettt xvii
Chapter 1 How to Use This Book 1

1.1 The Purposes of this Companion, 1
1.2 Finding the Meaning of Things, 3
1.3 Learning for the First Time, 3
1.4 Recalling, or Brushing Up, 4
1.5 IsItTrueThat...?,4
1.6 What Else Should One Know About...?,5
1.7 Reading the Book Sequentially, 5
1.8 The Companion as a Handbook, 6
1.9 The Companion as a Textbook, 6
Chapter 2 Sets and Subsets 8

2.1 Sets, Elements and Membership, 8
2.2 Difficulties in Describing Sets, 9
2.3 Describing Sets, 10
2.4 The Empty Set; Finite and Infinite Sets, 11
2.5 Equality of Sets; Subsets and Supersets, 12
2.6 The Power Set, 14
2.7 Indexing Sets, 15
Chapter 3 Basic Relations and Operations on Sets .......c.cccceecereceeccnece 16
3.1 Complements and Set Difference (Relative Complement), 16
3.2 Venn Diagrams, 17
3.3 The Law of Involution, 18
3.4 Union and Intersection, 19
3.5 The Subsets Generated by Two Sets, 21
3.6 Disjointness and Inclusion, 21
vii
viii CONTENTS
3.7 Union and Intersection Are Commutative and
Associative, 23
3.8 Both Distributive Laws Hold, 25
3.9 Idempotent Laws, 25
3.10 Laws of Complements, 25
3.11 Laws of Operating with ¢ and U, 26
3.12 DeMorgan’s Laws, 27
Chapter 4 The Principle of Duality; Further Properties of Sets............ 28
4.1 Absorption Laws, 28
4.2 The Principle of Duality, 29
4.3 Summary of Set Laws, 31
4.4 Laws of Set Difference (Relative Complement), 33
4.5 The Symmetric Difference, 33
4.6 Describing-Properties and Operations on Sets;
Constructing Set Expressions from Complicated
Descriptions, 35
4.7 Verifying Set-Expression Identities; Modified Venn
Diagrams, 39
4.8 Counting Arguments for Finite Sets, 43
4.9 The Principle of Inclusion and Exclusion, 45
4.10 Concatenation, Strings and Languages, 47
* 411 Infinite Intersection and Union, 53
* 412 Closure of Families of Sets, 55
* 413 Defining Sets by Closure, 57
** 4.14 Closure under Countable Unions and Intersections, 60
4.15 Set Product; More About Strings, 64
4.16 Set Closure Revisited; Yet More About Strings, 68
Chapter § Discrete Probability and Combinatorial Analysis ................ 72
5.1 Background: Theory vs. Practice, 72
5.2 Sample Points, Sample Space, Events and Probability, 73
5.3 Combinations of Events, 74
54 Probability Defined, 75
5.5 Basic Results, 76
*These sections are somewhat more demanding in mathematical maturity, or somewhat less
likely to be encountered by the computer scientist, or both.
**This section requires considerable mathematical maturity on the part of the reader. The topics
here may be encountered infrequently.