Introduction to

DISCRETE MATHEMATICS

with ISETL

Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Introduction to

DISCRETE MATHEMATICS

with ISETL

William E. Fenton Ed Dubinsky

.~ ~

Springer

William E. Fenton Department of Mathematics Bellarmine College Newburg Road Louisville, KY 40205 USA

Ed Dubinsky Department of Mathematics Purdue University West Lafayette, IN 47907 USA

Library of Congress Cataloging-in-Publication Data

Fenton, William (William E.) Introduction to discrete mathematics with ISETL I by William Fenton & Ed Dubinsky.

p. cm. Includes bibliographical references (p. - ) and index.

ISBN-13: 978-1-4612-8480-2 e-ISBN-13: 978-1-4612-4052-5 DOl: 10.1007/978-1-4612-4052-5

1. Mathematics. 2. Computer science-Mathematics. 3. ISETL (computer pro-gram language) I. Dubinsky. Ed. II. Title.

QA39.2.F445 1996 511.3'078--<ic20 96-8337

Printed on acid-free paper.

© 1996 Springer-Verlag New York Inc. Solkover reprint of the hardcover 1st edition 1996

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or here­after developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as un­derstood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production coordinated by Black Hole Publishing and managed by Terry Kornak; manufac­turing supervised by Joe Quatela.

Typeset by Black Hole Publishing, Berkeley, CA.

987654321

Contents

COMMENTS FOR THE INSTRUCTOR To the Student. . . . . . . .

1 NUMBERS AND PROGRAMS 1.1 The Basics of ISETL

Activities . . . . . . Discussion .....

Beginning with ISETL Some Syntax . . . . . . Familiar Sets of Numbers Decimal Representation Binary Representation . Sequences

Exercises 1.2 Divisibility ....

Activities Discussion

ISETLfuncs- Functions ISETL smaps- Functions. Sources of Functions Recursive Functions . Modular Arithmetic Prime Numbers .. Common Divisors . Common Multiples

Exercises · .. Overview of Chapter I ....

2 PROPOSITIONAL CALCULUS 2.1 Boolean Expressions

Activities · ...... Discussion ......

Constants and Variables Basic Operations. . . . Functions Using Boolean Values

Exercises · .. 2.2 Implication and Proof .

Activities · ..

xi xiv

1 1 1 1 1 3 3 5 5 6 6 9 9

12 12 13 14 15 15 16 18 20 21 24

27 27 27 29 29 30 32 33 34 34

vi CONTENTS

Discussion · .................. 36 Conditional Statements ....... 36 Variations of Conditional Statements 37 Direct Proof . . . . . . 38 Indirect Proof ..... 38 Proof by Contradiction 39

Exercises · .. 40 Overview of Chapter 2 41

3 SETS AND TUPLES 43 3.1 Defining Sets and Tuples 43

Activities · ... 43 Discussion · .. 46

Sets and their Elements 46 Tuples and their Elements . 48 Forming Sets and Tuples 48 Sequences ...... 50 Recursive Sequences 50

Exercises 51 3.2 Operations on Sets 53

Activities . 53 Discussion 55

Cardinality. 55 Subsets ... 55 Basic Combinations of Sets 57 De Morgan's Laws. 58 Cartesian Products . 59 Inclusion-Exclusion 59

Exercises 60 3.3 Counting Methods 62

Activities 62 Discussion 64

The Multiplication Principle. 64 Permutations . . . . . . . 65 Combinations ...... 66 The Pigeonhole Principle 67

Exercises · .. 68 Overview of Chapter 3 70

4 PREDICATE CALCULUS 73 4.1 Quantified Expressions 73

Activities · .. 73 Discussion · . 76

Existential and Universal Quantifiers 76

CONTENTS vii

Quantifying over Proposition Valued Functions­Existential . . . . . . . . . . . . . . . . . . .. 76 Quantifying over Proposition Valued Functions-Universal. . . . . . . . . . . . . . . . . 77 Negations ................ 78 Reasoning about Quantified Expressions 78

Exercises . . . . . 79 4.2 Multi-Level Quantification 84

Activities . . . . . 84 Discussion .... 87

Quantified Statements that Depend on a Variable 87 Two-Level Quantification . . . . . . . . . . 89 Negating Two-Level Quantifications .... 90 Reasoning about Two-Level Quantifications 91 Three-Level Quantification 92

Exercises . . . 92 Overview of Chapter 4 . . 96

5 RELATIONS AND GRAPHS 5.1 Relations and their Graphs

Activities . . . . . Discussion ....

Relations. Representing a Relation Properties of Relations More about Graphs .,

Exercises ............ . 5.2 Equivalence Relations and Graph Theory

Activities . . . . . . . . . . . . Discussion .......... .

Equivalence Relations Types of Graphs Subgraphs Planarity

Exercises ... Overview of Chapter 5

6 FUNCTIONS 6.1 Representing Functions .

Activities . . . . Discussion ...

Constructing Functions Functions as Expressions Functions as Sequences Functions as Tables . . .

97 97 97 99 99

100 101 102 103 106 106 107 107 109 109 III III 114

117 117 117 120 120 122 122 123

viii CONTENTS

Functions as Graphs . . . . The Process of a Function . Two Definitions

Exercises ... 6.2 Properties of Functions

Activities . . .

124 125 126 126 129 129

Discussion .. 132 Basic Properties 132 One-to-One Functions . 133 Combinations of Functions 135 Inverse Functions . . . . . 137 Rate of Growth for Functions 138

Exercises . . . 140 Overview of Chapter 6 . . . . 143

7 MATHEMATICAL INDUCTION 145 7.1 Understanding the Method 145

Activities . . . . . . . 145 Discussion ...... 148

Proposition-Valued Functions . 148 Eventually Constant Proposition-Valued Func-tions . . . . . . . . . . . . . . 148 Implication-Valued Functions. 149 Modus Ponens . . . . . 150 Coordinating the Steps

Exercises ...... . 7.2 Using Mathematical Induction

Activities . . . . . . . Discussion ..... .

Making Induction Proofs The Induction Principle Complete Induction . . The Binomial theorem .

Exercises ... Overview of Chapter 7

8 PARTIAL ORDERS Activities Discussion . . . . . . .

Order on a Set Diagrams of Posets Topological Sorting Sperner's Theorem.

Exercises ... Overview of Chapter 8 .. . . . . . .

151 151 153 153 154 154 156 156 157 158 160

163 163 164 164 165 166 167 168 170

CONTENTS ix

9 INFINITE SETS 173 Discussion ...... . . . . . . . . . . . . . . . . . . . 173

Sets of Equal Cardinality . . . . . . . . . . . . 173 Infinite Sets ................... 174 Countable Sets .................. 174 Uncountable Sets ................ 178 Ordering oflnfinite Sets . . . . . . . . . . . . . 179

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 180

APPENDIX 1: GETTING STARTED WITH ISETL 182 A. Working in the Execution Window ........... 182 B. Working with Files ................... 184 C. Using Directives .................... 185 D. Graphing in ISETL ................... 186

APPENDIX 2: SOME SPECIAL CODE 188

INDEX 191

INDEX OF FREQUENTLY USED SETS AND FUNCTIONS 194

COMMENTS FOR THE INSTRUCTOR

TEACHING A COURSE WITH THIS BOOK

This book is intended to support a constructivist approach to teaching (in the epistemological sense, not the mathematical). That is, it can be used in an under­graduate discrete mathematics course or in a "bridge" course to help create an environment in which students construct for themselves mathematical concepts appropriate to understanding and solving problems. Of course, the pedagogical ideas on which the book is based do not appear explicitly in its text, but rather are implicit in its structure and content.

The mathematical ideas in our text are not presented in a completed and pol­ished form, adhering to a strict logical sequence, but are presented roughly and circularly, with the student responsible for trying to straighten things out. The student is given considerable help in making mathematical constructions to use in making sense out of the material. This help comes from a combination of com­puter activities, leading questions, and a conversational writing style.

It should be noted that although each learning cycle begins with activities pre­ceding a discussion of some of the mathematical ideas on which they are based, the students are not expected to discover all of the mathematics for themselves. In fact, since the main purpose of the activities is to establish an experiential base for subsequent learning, students who spend considerable time and effort work­ing on the activities will reap benefits whether they have discovered the "right" answers or not. Definitions, theorems, proofs and summaries are presented only after the student has had a chance to work with the ideas related to the concepts to be formalized.

In teaching courses based on this book and on similar texts in other subjects, we find that our approach appears to be extremely effective for most students, bring­ing them a stronger understanding of mathematical ideas than one would think possible from experience with traditional pedagogy. The exercise set is strong enough to challenge superior students and whet their appetite for more mathemat­ics. And we find that students of all levels who go through our courses develop a positive attitude towards mathematical abstraction and mathematics in general.

Finally, before describing the structure of the book, we should point out that this text is only part of the course. The authors are preparing a package to aid instruc­tors in using this approach. This package will include discussion of the activities and their solutions, samples of assignments, class lesson plans, sample exams, and information on dividing students into teams. This material will be available

XII COMMENTS FOR THE INSTRUCTOR

from the authors. An ISETL user's manual by Jennie Dautermann, entitled Using ISETL 3.0: A Language for Learning Mathematics, is available from West Pub­lishing (ISBN 0-314-01326-1 for the Macintosh version, and 0-314-01327-X for the DOS version).

THE A.C.E. CYCLE

The text is divided into sections, each intended to be covered by an average class in about a week. A section consists of a set of Activities, Class discussion material, and a set of Exercises.

Activities

These tasks present problems which require students, working in teams, to write ISETL code representing mathematical constructs that can be used to solve the problems. Often an activity will require use of mathematics not yet covered in the text. The student is expected to try to discover the mathematics or sometimes just make guesses, possibly reading ahead in the text for clues or explanations.

Class Discussion Material

These portions contain some explanations, some completed mathematics, and many questions, all taking place under the assumption that the student has already spent considerable time and effort on the activities related to the same topics. Our experience indicates that, with this background, students can relate much more meaningfully to formal definitions and theorems. Each unanswered question in the text is either answered later in the book or repeated as an explicit problem in the exercises. Our way of using this discussion material in a course is to have the students work together in teams during class to solve paper and pencil problems, mainly suggested by the open questions in the text. This largely replaces lectures, which occur only as summations after the students have had a chance, through the activities and discussions, to understand the material.

Exercises

There are two kinds of exercises in this text. Some exercises are relatively tra­ditional and are used to reinforce and solidify the ideas that the students have constructed up to this point. They occasionally introduce preliminary versions of topics that will be considered later. Other exercises, especially in the chapters on Predicate Calculus, Functions, and Mathematical Induction, have been taken from our research into learning and represent problems we have used to understand the difference between students who are constructing useful mathematical ideas and those who are not.

COMMENTS FOR THE INSTRUCTOR xiii

Obtaining ISETL

A user's manual is available for ISETL, in separate versions for DOS or Mac­intosh. These come from the West Publishing Company and a disk is included. ISETL is also available on the World Wide Web at

http://www.mth.cmich.edu/faculty/mathewsllinks.htrnl

Another option is to contact either one of the authors. A Windows version ofISETL has recently been developed by John Kirchmeyer

of Mount Union College. Version 1.2 is a true Windows application with pull­down menus, a button bar, and mouse abilities. As of May 1996 ISETL for Win­dows lacks graphics capabilities, but work continues and graphics are expected to be part of the next version. Information and the most current version of ISETL for Windows can be obtained from the WWW at

http://csis03 .muc.edu/isetlw /istelw.html

Please note that ISETL is a free software. We encourage instructors to make it freely available to students for their own computers.

COVERING THE COURSE MATERIAL

Though the teaching method supported by this book is novel, the selection of material is fairly standard. The text can be used for an introductory lower level college course in discrete mathematics or, perhaps with some supplementary ma­terial provided by the instructor, for a higher level college course intended to pro­vide a bridge to the study of more advanced mathematical topics. The first chap­ter provides the student with the opportunity to learn how to program in ISETL while simultaneously learning or reviewing certain basic material. Chapters 2-8 cover standard topics in elementary discrete mathematics. The instructor should be aware that the activities often rely on activities from earlier sections, so care is called for when assigning problems.

Chapter 9-which we are introducing in this text as an informal experiment­represents a departure from our methodology. There is no question that some students do not need the special kind of pedagogy and interaction with computers that this text supports. These are individuals who, we believe, have special talent in this subject. We think of them as the "natural athletes" of mathematics. Most students, however, need some assistance, some scaffolding to help them construct mathematical concepts; such is the main aim of this text and of the course of which it is a part. Whether students in general will always need the kind of support that a constructivist pedagogical approach tries to provide is, for us, an open question. It might be that some students, after experiencing a course such as this, will develop the ability to construct mathematical concepts in their minds by listening to a presentation of the material, or by reading about it in a text-plus, of course, by working problems. Chapter 9 is written in a completely traditional manner,

xiv COMMENTS FOR THE INSTRUCTOR

and we recommend that it be presented in class by the instructor explaining the material and discussing it with students. The question it raises is whether some students, at the end of this course, will be able to handle mathematics presented in a traditional manner.

Returning to our constructivist pedagogy, the experiences of the many faculty who have implemented this and similar texts in courses using our pedagogical methods, indicate that this approach can provide the opportunity for students to gain a much deeper understanding of the concepts of mathematics than may be possible in a course taught in a more traditional manner. We hope your experience will be similar.

TO THE STUDENT

In this book we ask a lot of questions. You should ask a lot of questions too, and the first one might be: What is Discrete Mathematics? Most of the mathemat­ics you have studied up to now has been continuous mathematics, which deals with continuous sets such as intervals, the number line, or the coordinate plane. These concepts will be useful for studying discrete mathematics, but the empha­sis will now be on sets with discrete elements, elements which are distinct and disconnected from each other. Sets with only a finite number of elements are au­tomatically discrete sets, though the infinitely many integers form a discrete set too. A large portion of discrete mathematics is concerned with finite sets. Finite sets sound as though they are easier to understand than infinite sets, but this can be misleading. After all, the size of the U.S. national debt is a finite number; do you think you understand it?

A natural question about a finite set is: How big is it? This simple query can be surprisingly difficult to answer. Think about the following set:

L:= the set of positive three-digit numbers in which the first digit is less than the other two digits

This has to be a finite set, since it contains just some of the positive three-digit numbers and there are only 900 of those. (Do you see why there are only 900?) But how many elements does L have? Counting elements in a set is part of discrete mathematics.

Discrete mathematics also looks at mathematical structures that can be built on discrete sets. We could ask how the elements could be compared to each other. The point [5,3] in the coordinate plane can be either "bigger" or "smaller" than the point [2,6]; it depends on what you mean by "bigger." The points could even be considered "equal"; can you think of a way to do this? Another aspect of math­em&tical structure asks how to combine elements to make new elements and how these combining operations work. For instance, it is possible to do interesting arithmetic using only single-digit numbers. This arithmetic is not what you are used to, since fi + 9 has to give a single-digit answer. Sets can be combined to

COMMENTS FOR THE INSTRUCTOR xv

fonn new sets; the size of these combinations depends on how the sets are merged. Sentences can be combined to fonn new sentences; whether the new sentence is true or false hinges on how the combination is made.

A second question you might ask about this book is: Why use computers? An historical reason is that many problems in discrete mathematics require a lot of computation. The computer has made great contributions to the subject by help­ing to solve such problems. Further, computer science has raised many problems which can be analyzed with techniques from discrete mathematics. So the two areas have strong ties. But we think there is a better reason for using a computer. It is an enonnously valuable tool for learning the concepts of discrete mathemat­ics. Interaction with a computer will help you build mathematical structures in your mind, which will strengthen your ability to analyze problems in a mathe­matical way. Writing the precise instructions that a computer needs will help to clarify your understanding of a problem. When you think through the steps that a computer has to use to calculate a result, you will better comprehend how the mathematics operates in the problem.

The computer-based problems in this text use a programming language called ISETL, which stands for Interactive SET Language. Much of this language looks like standard mathematical notation, so while you learn ISETL you will actually be learning mathematics. There are a few syntax details to get used to, such as putting a semicolon at the end of every line; these will come to you fairly quickly. There are always frustrations when working with a computer, but after a short time, syntax will be a minor point. More often, the difficulties will be with un­derstanding the underlying mathematics. This is a signal to think harder about the concepts and about how you are investigating the problem.

If you have looked through the book already, you have probably noticed that each section starts with activity problems. Working on the activities will help you build a foundation for a strong understanding of the concepts in that section. Feel free to guess at possible solutions and then test your guesses. If you need help with ISETL, check Appendix 1 or ask your instructor if a copy of the manual is available. You should certainly read the text of the section while you are work­ing on the activities, or even beforehand. The written discussions talk about the concepts used in the activities, and sometimes there are hints on how to solve these problems. So reading ahead is a good idea. At the end of each chapter is an overview of the topics included in that chapter. Sometimes the activities ask you to develop fonnulas for various concepts. We suggest that you write these fonnu­las on the overview pages and also fill in the important definitions. By doing this, you will create your own reference guide.

THE COMPUTER

There are certain basic things you will have to know about the computers that you will be using. The answers to the following questions are different from one computer to another, so we cannot respond to them here:

xvi COMMENTS FOR THE INSTRUCTOR

• How do you tum the computer on or off?

• How do you enter information? Keyboard, mouse, disks?

• How do you give commands? Menus, icons?

• How are the files organized?

• How do you save files or remove them?

• How do you get into ISETL?

• How do you print something?

• How do you edit the worksheet on the screen? Add, delete, or change text?

Some of these questions are dealt with in Appendix 1. Your instructor can help you with the others. It looks like a lot to learn all at once, but after a short time these will be routine issues, and then you can concentrate on the mathematics.

By working the problems in this book you will learn ISETL. But if you wish to learn more, you should get a copy of the manual, Using ISETL 3.0: A Language for Learning Mathematics by Jennie Dautermann. This is available from West Publishing in either a DOS or Macintosh version.

Bill Fenton / Bellarmine College Ed Dubinsky / Purdue University