Introduction to Discrete Mathematics with ISETL

205 Pages·1996·6.119 MB·English

by  William E. Fenton, Ed Dubinsky (auth.)

Introduction to
DISCRETE
MATHEMATICS
with ISETL
Springer
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Introduction to
DISCRETE
MATHEMATICS
with ISETL
William E. Fenton
Ed Dubinsky
.~
~
Springer
William E. Fenton Ed Dubinsky
Department of Mathematics Department of Mathematics
Bellarmine College Purdue University
Newburg Road West Lafayette, IN 47907
Louisville, KY 40205 USA
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
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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
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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 un
derstood 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; manufac
turing supervised by Joe Quatela.
Typeset by Black Hole Publishing, Berkeley, CA.
987654321
Contents
COMMENTS FOR THE INSTRUCTOR xi
To the Student. . . . . . . . xiv
1 NUMBERS AND PROGRAMS 1
1.1 The Basics of ISETL 1
Activities . . . . . . 1
Discussion ..... 1
Beginning with ISETL 1
Some Syntax . . . . . . 3
Familiar Sets of Numbers 3
Decimal Representation 5
Binary Representation . 5
Sequences 6
Exercises 6
1.2 Divisibility .... 9
Activities 9
Discussion 12
ISETLfuncs-Functions 12
ISETL smaps-Functions. 13
Sources of Functions 14
Recursive Functions . 15
Modular Arithmetic 15
Prime Numbers .. 16
Common Divisors . 18
Common Multiples 20
Exercises · .. 21
Overview of Chapter I .... 24
2 PROPOSITIONAL CALCULUS 27
2.1 Boolean Expressions 27
Activities · ...... 27
Discussion ...... 29
Constants and Variables 29
Basic Operations. . . . 30
Functions Using Boolean Values 32
Exercises · .. 33
2.2 Implication and Proof . 34
Activities · .. 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 Functions
Existential . . . . . . . . . . . . . . . . . . .. 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 97
5.1 Relations and their Graphs 97
Activities . . . . . 97
Discussion .... 99
Relations. 99
Representing a Relation 100
Properties of Relations 101
More about Graphs ., 102
Exercises ............ . 103
5.2 Equivalence Relations and Graph Theory 106
Activities . . . . . . . . . . . . 106
Discussion .......... . 107
Equivalence Relations 107
Types of Graphs 109
Subgraphs 109
Planarity III
Exercises ... III
Overview of Chapter 5 114
6 FUNCTIONS 117
6.1 Representing Functions . 117
Activities . . . . 117
Discussion ... 120
Constructing Functions 120
Functions as Expressions 122
Functions as Sequences 122
Functions as Tables . . . 123
viii CONTENTS
Functions as Graphs . . . . 124
The Process of a Function . 125
Two Definitions 126
Exercises ... 126
6.2 Properties of Functions 129
Activities . . . 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 151
Exercises ...... . 151
7.2 Using Mathematical Induction 153
Activities . . . . . . . 153
Discussion ..... . 154
Making Induction Proofs 154
The Induction Principle 156
Complete Induction . . 156
The Binomial theorem . 157
Exercises ... 158
Overview of Chapter 7 160
8 PARTIAL ORDERS 163
Activities 163
Discussion . . . . . . . 164
Order on a Set 164
Diagrams of Posets 165
Topological Sorting 166
Sperner's Theorem. 167
Exercises ... 168
Overview of Chapter 8 .. . . . . . . 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 under
graduate 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 pol
ished 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 com
puter activities, leading questions, and a conversational writing style.
It should be noted that although each learning cycle begins with activities pre
ceding 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 work
ing 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, bring
ing 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 mathemat
ics. 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 instruc
tors 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