A Spiral Workbook for Discrete Mathematics

307 Pages·2015·1.856 MB·English

by  Harris Kwong

A Spiral Workbook
for
Discrete Mathematics
Harris Kwong
Dept. of Math. Sci.
SUNY Fredonia
Open SUNY Textbooks
2015
(cid:13)c2015 Harris Kwong
ISBN:978-1-942341-16-1
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About the Book
A Spiral Workbook for Discrete Mathematics covers the standard topics in a sophomore-level
course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions,
relations,andelementarycombinatorics,withanemphasisonmotivation. Thetextexplainsand
clarifies the unwritten conventions in mathematics, and guides the students through a detailed
discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises
help students understand a concept soon after learning it. The text adopts a spiral approach:
many topics are revisited multiple times, sometimes from a different perspective or at a higher
level of complexity, in order to slowly develop the student’s problem-solving and writing skills.
About the Author
Harris Kwong is a mathematics professor at SUNY Fredonia. He was born and raised in Hong
Kong. After finishing high school there, he came to the United States to further his education.
He received his B.S. and M.S. degrees from the University of Michigan, and Ph.D. from the
University of Pennsylvania. His research focuses on combinatorics, number theory, and graph
theory. Hisworkappearsinmanyinternationalmathematicsjournals. Besidesresearcharticles,
he also contributes frequently to the problems and solutions sections of Mathematics Monthly,
Mathematics Magazine, College Journal of Mathematics, and Fibonacci Quarterly. He gives
thanks and praises to God for his success.
About Open SUNY Textbooks
Open SUNY Textbooks is an open access textbook publishing initiative established by State
University of New York libraries and supported by SUNY Innovative Instruction Technology
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and establishing a community of practice among libraries. Participating libraries in the 2012-
2013 pilot include SUNY Geneseo, College at Brockport, College of Environmental Science and
Forestry, SUNY Fredonia, Upstate Medical University, and University at Buffalo, with support
from other SUNY libraries and SUNY Press. To date, the project has published 10 open
textbooks. More information can be found at http://textbooks.opensuny.org.

Preface
Therearemanydiscretemathematicstextbooksavailable,sowhydidIdecidetoinvestmytime
and energy to work on something that perhaps only I myself would appreciate?
Mathematical writings are full of jargon and conventions that, without proper guidance, are
difficult for beginners to follow. In the past, students were expected to pick them up along the
way on their own. Those who failed to do so would be left behind. Looking back, I consider
myself lucky. It was by God’s grace that I survived all those years. Now, when I teach a
mathematical concept, I discuss its motivation, explain why it is important, and provide a lot
of examples. I dissect the proofs thoroughly to make sure everyone understands them. In brief,
I want to show my students how to analyze mathematical problems.
Most textbooks typically hide all these details. They only show you the final polished
products. By training, mathematicians love short and elegant proofs. This is reflected in their
own writing. Yes, the results are beautiful, but it is a mystery how mathematicians come up
with such ideas. I want a textbook that discusses mathematical concepts in greater detail. I
want to teach my students how to read and write mathematical arguments. Since I could not
find a textbook that suited my needs, I started writing lecture notes to supplement the main
text. Marginal notes, hands-on exercises, summaries, and section exercises were subsequently
added at different stages. The lecture notes have evolved into a full-length text.
Discrete mathematics is a rich subject, full of many interesting topics. Often, it is taught to
both mathematics and computer science majors. Due to the limit in space, this text addresses
mainly the needs of the mathematics majors. Consequently, we will concentrate on logic and
proof techniques, and apply them to sets, basic number theory, and functions. In the last two
chapters,wediscussrelationsandcombinatorics,asmanystudentswillfindthemusefulinother
courses.
Since the intended audience of the text is mathematics majors, I use a number of examples
from calculus. By design, I hope this can help the students review what they have learned, and
see that discrete mathematics forms the foundation of many mathematical arguments.
Discrete mathematics is often a required course in computer science. I find it hard and
unjust to serve two different groups of students in the same textbook. Although this text could
be used in a typical first semester discrete mathematics class for the computer science majors,
they need to consult another text for the second semester course. Here are two that serve this
purpose well:
• Alan Doerr and Kenneth Levasseur, Applied Discrete Structures.
• Miguel A. Lerma, Notes on Discrete Mathematics.
Both are available on-line.
WhydoIcallthisaworkbook? Therearemanyhands-onexercisesdesignedtohelpstudents
understandanewconceptbeforetheymoveontothenext. IbelievethetitleWorkbook reflects
the nature of the book, because I expect the students to work on the hands-on exercises. But
why spiral? Because the pedagogy is inspired by the spiral method. The idea is to revisit
some themes and results several times throughout the course and each time further deepen
your understanding. You will find some problems pop up more than once, and are solved in a
different way each time. In other instances, a concept you learned earlier will be viewed from a
new perspective, thus adding a new dimension to it.
vi Preface
I am indebted to the anonymous reviewers, whose numerous valuable comments helped to
shape the workbook in its current form. I would also like to express my great appreciation to
ScottRichmondofReedLibraryattheStateUniversityofNewYorkatFredonia,whoprovided
many helpful suggestions and editorial assistance.
ThereasonIdevelopedthisworkbookistohelpstudentslearndiscretemathematics. Ifthis
workbook proves to be a failure, I am the one to blame. If you find this workbook serves its
intended purposes, I give all the glory to God, in whom I believe and trust.
Harris Kwong
April 21, 2015
Contents
Preface v
1 An Introduction 1
1.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Suggestions to Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 How to Read and Write Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Proving Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Logic 9
2.1 Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Conjunctions and Disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Biconditional Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Logical Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Logical Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Proof Techniques 43
3.1 An Introduction to Proof Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Direct Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Indirect Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Mathematical Induction: An Introduction . . . . . . . . . . . . . . . . . . . . . . 58
3.5 More on Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Mathematical Induction: The Strong Form . . . . . . . . . . . . . . . . . . . . . 73
4 Sets 81
4.1 An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Subsets and Power Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 Unions and Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.5 Index Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Basic Number Theory 117
5.1 The Principle of Well-Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.4 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5 More on GCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.6 Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.7 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
viii CONTENTS
6 Functions 157
6.1 Functions: An Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2 Definition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.3 One-to-One Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4 Onto Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.5 Properties of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.6 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.7 Composite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7 Relations 197
7.1 Definition of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Properties of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.3 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4 Partial and Total Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8 Combinatorics 221
8.1 What is Combinatorics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.2 Addition and Multiplication Principles . . . . . . . . . . . . . . . . . . . . . . . . 222
8.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.4 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.5 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
A Solutions to Hands-On Exercises 251
B Answers to Selected Exercises 279
Index 296
Chapter 1
An Introduction
1.1 An Overview
What is discrete mathematics? Roughly speaking, it is the study of discrete objects. Here,
discrete means “containing distinct or unconnected elements.” Examples include:
• Determining whether a mathematical argument is logically correct.
• Studying the relationship between finite sets.
• Counting the number of ways to arrange objects in a certain pattern.
• Analyzing processes that involve a finite number of steps.
Here are a few reasons why we study discrete mathematics:
• To develop our ability to understand and create mathematical arguments.
• To provide the mathematical foundation for advanced mathematics and computer science
courses.
In this text, we will cover these five topics:
1. Logic and Proof Techniques. Logic allows us to determine if a certain argument is
valid. We will also learn several basic proof techniques.
2. Sets. We study the fundamental properties of sets, and we will use the proof techniques
we learned to prove important results in set theory.
3. Basic Number Theory. Numbertheoryisoneoftheoldestbranchesofmathematics;it
studies properties of integers. Again, we will use the proof techniques we learned to prove
some basic facts in number theory.
4. Relations and Functions. Relations and functions describe the relationship between
the elements from two sets. They play a key role in mathematics.
5. Combinatorics. Combinatorics studies the arrangement of objects. For instance, one
may ask, in how many ways can we form a five-letter word. It is used in many disciplines
beyond mathematics.
Allofthesetopicsarecrucialinthedevelopmentofyourmathematicalmaturity. Theimportance
ofsomeoftheseconceptsmaynotbeapparentatthebeginning. Astimegoeson,youwillslowly
understand why we cover such topics. In fact, you may not fully appreciate the subjects until
you start taking advanced courses in mathematics.
This is a very challenging course partly because of its intensity. We have to cover many
topics that appear totally unrelated at first. This is also the first time many students have to
studymathematicsindepth. Youwillbeaskedtowriteupyourmathematicalargumentclearly,
precisely, and rigorously, which is a new experience for most of you.