A Mathematical Foundation for Computer Science, Preliminary Edition

364 Pages·2019·218.843 MB·English

by  David Mix Barrington

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A Mathematical Foundation
for Computer Science
PRELIMINARY EDITION
David Mix Barrington
Kendall Hunt
publishing co mp any
All chapter heading quotes are taken from Monty Python's Flying Circus: All the Words
(Volumes 1 and 2) (New York: Pantheon Books, 1989) and Monty Python and The Holy Grail
(Book) [M0nti Pyth0n ik den H0lie Grailen (B0k)J (New York, Methuen Inc., 1979).
Excursion 1.11 uses text from Through the Looking Glass, and What Alice Found There by
Lewis Carroll (London, McMillan, 1871) and Fox in Socks by Dr. Seuss (New York, Random
House, 1965).
Problems 2.6.2 and 2.6.3 use text from The Number oft he Beast by Robert A. Heinlein (New
York: Fawcett, 1980).
There are many references in the text to Godel, Escher, Bach: An Eternal Golden Braid by
Douglas R. Hofstadter (New York: Basic Books, 1979).
Cover image of Stalker Castle, Scotland, by Frank Parolek © Shutterstock, Inc.
Kendall Hunt
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www.kendallhunt.com
Send all inquiries to:
4050 Westmark Drive
Dubuque, IA 52004-1840
Copyright© 2019 by Kendall Hunt Publishing Company
ISBN 978-1-7924-0564-8
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the copyright owner.
Published in the United States of America
PRELIMINARY EDITION CONTENTS
Chapter 1: Sets, Propositions, and Predicates 1-1
1.1: Sets 1-2
1.2: Strings and String Operations 1-11
1.3: Excursion: What is a Proof? 1-20
1.4: Propositions and Boolean Operations 1-24
1.5: Set Operations and Propositions About Sets 1-34
1.6: Truth-Table Proofs 1-45
1.7: Rules for Propositional Proofs 1-52
1.8: Propositional Proof Strategies 1-59
1.9: Excursion: A Murder Mystery 1-65
1. 10: Predicates 1-68
1.11: Excursion: Translating Predicates 1-75
Glossary for Chapter 1 1-78
Chapter 2: Quantifiers and Predicate Calculus 2-1
2.1: Relations 2-2
2.2: Excursion: Relational Databases 2-8
2.3: Quantifiers 2-10
2.4: Excursion: Translating Quantifiers 2-17
2.5: Operations on Languages 2-19
2.6: Proofs With Quantifiers 2-25
2.7: Excursion: Practicing Proofs 2-32
2.8: Properties of Binary Relations 2-34
2.9: Functions 2-41
2.10: Partial Orders 2-48
2.11: Equivalence Relations 2-55
Glossary for Chapter 2 2-62
Chapter 3: Number Theory 3-1
3 .1: Divisibility and Primes 3-2
3.2: Excursion: Playing With Numbers 3-11
3.3: Modular Arithmetic 3-14
3.4: There are Infinitely Many Primes 3-23
3.5: The Chinese Remainder Theorem 3-28
3.6: The Fundamental Theorem of Arithmetic 3-35
3.7: Excursion: Expressing Predicates in Number Theory 3-43
3.8: The Ring of Congruence Classes 3-46
3.9: Finite Fields and Modular Exponentiation 3-52
3.10: Excursion: Certificates of Primality 3-58
3.11: The RSA Cryptosystem 3-61
Glossary for Chapter 3 3-71
Chapter 4: Recursion and Proof by Induction 4-1
4.1: Recursive Definition 4-2
4.2: Excursion: Recursive Algorithms 4-10
4.3: Proof By Induction for Naturals 4-13
4.4: Variations on Induction for Naturals 4-20
4.5: Excursion: Fibonacci Numbers 4-27
4.6: Proving the Basic Facts of Arithmetic 4-30
4.7: Recursive Definition for Strings 4-37
4.8: Excursion: Naturals and Strings 4-45
4.9: Graphs and Paths 4-47
4.10: Trees and Lisp Lists 4-56
4.11: Induction for Problem Solving 4-66
Glossary for Chapter 4 4-75
S.1: Solutions to Exercises from Chapter 1 S-1
S.2: Solutions to Exercises from Chapter 2 S-16
S.3: Solutions to Exercises from Chapter 3 S-32
S.4: Solutions to Exercises from Chapter 4 S-48
FULL VERSION CONTENTS
Chapter 1: Sets, Propositions and Predicates
Chapter 2: Quantifiers and Predicate Calculus
Chapter 3: Number Theory
Chapter 4: Recursion and Proof By Induction
Chapter 5: Regular Expressions and Other Recursive Systems
Chapter 6: Fundamental Counting Problems
Chapter 7: Further Topics in Combinatorics
Chapter 8: Graphs
Chapter 9: Trees and Searching
Chapter 10: Discrete Probability
Chapter 11: Reasoning About Uncertainty
Chapter 12: Markov Processes and Classical Games
Chapter 13: Information Theory
Chapter 14: Finite-State Machines
Chapter 15: A Brief Tour of Formal Language Theory

AUTHOR'S NOTE
• This Preliminary Edition contains the first four of the entire
book's fifteen chapters, which form the first half of the text for
COMPSCI 250 at UMass Amherst in Fall 2019. The final
version will include all eight chapters used in 250 and seven
others which could be used in COMPSCI 240.
• Each chapter has eight ordinary sections and three Excursions.
In COMPSCI 250 the 50-minute lectures cover one or
sometimes two ordinary sections, and Excursions are used for
team problem-solving sessions in the weekly discussion
sections. Each ordinary section contains ten Exercises (with
solutions in the back) and ten Problems (suitable for homework
assignments).
Grateful Thanks to:
• First and foremost, my wife Jessica and daughter Julia.
• The many teachers who helped form me as a mathematician,
including David Cox, the late Jim Mauldon, and especially
Mark Kidwell at Amherst College, Adrian Mathias at
Cambridge University, and Mike Sipser at M.I.T.
• Colleagues at UMass who helped form me as a teacher and
COMPSCI 250 as a course, including Amy Rosenberg, Neil
Immerman, Hava Siegelmann, Marius Minea, and dozens of
graduate and undergraduate teaching assistants.
• My students in this and other courses.
• Emma Strubell, who made most of the diagrams in Chapters 1
and 2.
• Bev Kraus and Lenell Wyman at Kendall Hunt.

Chapter 1: Sets, Propositions, and Predicates
"I came here for a good argument. "
"No you didn't, you came here for an argument. "
"Well, an argument's not the same as contradiction."
"It can be."
"No it can't. An argument is a connected series of statements intended to establish
a definite proposition."
"No it isn't. "
"Yes it is. It isn't just contradiction."
"Look, if I argue with you I must take up a contrary position. "
"But it isn't just saying 'No, it isn't'."
"Yes it is."
"No it isn't, argument is an intellectual process ... contradiction is just the automatic
gainsaying of anything the other person says."
"No it isn't. "
Our overall goal is to become familiar with a variety of mathematical objects, and learn to both
make and prove precise statements about them. In this opening chapter, we will define some of
the objects, develop the necessary language and vocabulary to make these statements, and begin
to see how to prove them. More specifically, we will:
• Give definitions and examples of the most basic objects of discrete mathematics: sets, strings,
and formal languages.
• Define propositions (boolean variables) and consider the propositional calculus, a method of
making, manipulating, and proving combinations of propositions.
• Define predicates, functions that take arguments of some type and return propositions. Pred
icates can be used to model more complicated English statements within the propositional
calculus.
1-1