Book of Proof

313 Pages·2013·1.35 MB·English

by  Richard Hammack

Book of Proof
Richard Hammack
Virginia Commonwealth University
RichardHammack(publisher)
DepartmentofMathematics&AppliedMathematics
P.O.Box842014
VirginiaCommonwealthUniversity
Richmond,Virginia,23284
BookofProof
Edition2.2
©2013byRichardHammack
This work is licensed under the Creative Commons Attribution-No Derivative Works 3.0
License
Typesetin11ptTEXGyreScholausingPDFLATEX
To my students
Contents
Preface vii
Introduction viii
I Fundamentals
1. Sets 3
1.1. IntroductiontoSets 3
1.2. TheCartesianProduct 8
1.3. Subsets 11
1.4. PowerSets 14
1.5. Union,Intersection,Difference 17
1.6. Complement 19
1.7. VennDiagrams 21
1.8. IndexedSets 24
1.9. SetsthatAreNumberSystems 29
1.10. Russell’sParadox 31
2. Logic 33
2.1. Statements 34
2.2. And,Or,Not 38
2.3. ConditionalStatements 41
2.4. BiconditionalStatements 44
2.5. TruthTablesforStatements 46
2.6. LogicalEquivalence 49
2.7. Quantifiers 51
2.8. MoreonConditionalStatements 54
2.9. TranslatingEnglishtoSymbolicLogic 55
2.10. NegatingStatements 57
2.11. LogicalInference 61
2.12. AnImportantNote 62
3. Counting 63
3.1. CountingLists 63
3.2. Factorials 70
3.3. CountingSubsets 73
3.4. Pascal’sTriangleandtheBinomialTheorem 78
3.5. Inclusion-Exclusion 81
v
II How to Prove Conditional Statements
4. Direct Proof 87
4.1. Theorems 87
4.2. Definitions 89
4.3. DirectProof 92
4.4. UsingCases 98
4.5. TreatingSimilarCases 99
5. Contrapositive Proof 102
5.1. ContrapositiveProof 102
5.2. CongruenceofIntegers 105
5.3. MathematicalWriting 107
6. Proof by Contradiction 111
6.1. ProvingStatementswithContradiction 112
6.2. ProvingConditionalStatementsbyContradiction 115
6.3. CombiningTechniques 116
6.4. SomeWordsofAdvice 117
III More on Proof
7. Proving Non-Conditional Statements 121
7.1. If-and-Only-IfProof 121
7.2. EquivalentStatements 123
7.3. ExistenceProofs;ExistenceandUniquenessProofs 124
7.4. ConstructiveVersusNon-ConstructiveProofs 128
8. Proofs Involving Sets 131
8.1. HowtoProve a∈A 131
8.2. HowtoProve A⊆B 133
8.3. HowtoProve A=B 136
8.4. Examples: PerfectNumbers 139
9. Disproof 146
9.1. Counterexamples 148
9.2. DisprovingExistenceStatements 150
9.3. DisproofbyContradiction 152
10. Mathematical Induction 154
10.1. ProofbyStrongInduction 161
10.2. ProofbySmallestCounterexample 165
10.3. FibonacciNumbers 167
vi
IV Relations, Functions and Cardinality
11. Relations 175
11.1. PropertiesofRelations 179
11.2. EquivalenceRelations 184
11.3. EquivalenceClassesandPartitions 188
11.4. TheIntegersModulo n 191
11.5. RelationsBetweenSets 194
12. Functions 196
12.1. Functions 196
12.2. InjectiveandSurjectiveFunctions 201
12.3. ThePigeonholePrinciple 205
12.4. Composition 208
12.5. InverseFunctions 211
12.6. ImageandPreimage 214
13. Cardinality of Sets 217
13.1. SetswithEqualCardinalities 217
13.2. CountableandUncountableSets 223
13.3. ComparingCardinalities 228
13.4. TheCantor-Bernstein-SchröederTheorem 232
Conclusion 239
Solutions 240
Index 301
Preface
I
n writing this book I have been motivated by the desire to create a
high-quality textbook that costs almost nothing.
The book is available on my web page for free, and the paperback
version (produced through an on-demand press) costs considerably less
than comparable traditional textbooks. Any revisions or new editions
will be issued solely for the purpose of correcting mistakes and clarifying
exposition. New exercises may be added, but the existing ones will not be
unnecessarily changed or renumbered.
This text is an expansion and refinement of lecture notes I developed
while teaching proofs courses over the past fourteen years at Virginia
Commonwealth University (a large state university) and Randolph-Macon
College (a small liberal arts college). I found the needs of these two
audiences to be nearly identical, and I wrote this book for them. But I am
mindful of a larger audience. I believe this book is suitable for almost any
undergraduate mathematics program.
This second edition incorporates many minor corrections and additions
that were suggested by readers around the world. In addition, several
new examples and exercises have been added, and a section on the Cantor-
Bernstein-Schröeder theorem has been added to Chapter 13.
Richard Hammack Richmond, Virginia
May 25, 2013
Introduction
T
his is a book about how to prove theorems.
Until this point in your education, mathematics has probably been
presented as a primarily computational discipline. You have learned to
solve equations, compute derivatives and integrals, multiply matrices
and find determinants; and you have seen how these things can answer
practical questions about the real world. In this setting, your primary goal
in using mathematics has been to compute answers.
But there is another side of mathematics that is more theoretical than
computational. Here the primary goal is to understand mathematical
structures, to prove mathematical statements, and even to invent or
discover new mathematical theorems and theories. The mathematical
techniques and procedures that you have learned and used up until now
are founded on this theoretical side of mathematics. For example, in
computing the area under a curve, you use the fundamental theorem of
calculus. It is because this theorem is true that your answer is correct.
However, in learning calculus you were probably far more concerned with
how that theorem could be applied than in understanding why it is true.
But how do we know it is true? How can we convince ourselves or others
of its validity? Questions of this nature belong to the theoretical realm of
mathematics. This book is an introduction to that realm.
This book will initiate you into an esoteric world. You will learn and
apply the methods of thought that mathematicians use to verify theorems,
explore mathematical truth and create new mathematical theories. This
will prepare you for advanced mathematics courses, for you will be better
able to understand proofs, write your own proofs and think critically and
inquisitively about mathematics.
ix
The book is organized into four parts, as outlined below.
PART I Fundamentals
• Chapter 1: Sets
• Chapter 2: Logic
• Chapter 3: Counting
Chapters1and2layoutthelanguageandconventionsusedinalladvanced
mathematics. Setsarefundamentalbecauseeverymathematicalstructure,
object or entity can be described as a set. Logic is fundamental because it
allowsustounderstandthemeaningsofstatements, todeduceinformation
about mathematical structures and to uncover further structures. All
subsequent chapters will build on these first two chapters. Chapter 3
is included partly because its topics are central to many branches of
mathematics,butalsobecauseitisasourceofmanyexamplesandexercises
thatoccurthroughoutthebook. (However,thecourseinstructormaychoose
to omit Chapter 3.)
PART II Proving Conditional Statements
• Chapter 4: Direct Proof
• Chapter 5: Contrapositive Proof
• Chapter 6: Proof by Contradiction
Chapters 4 through 6 are concerned with three main techniques used for
proving theorems that have the “conditional” form “If P, then Q.”
PART III More on Proof
• Chapter 7: Proving Non-Conditional Statements
• Chapter 8: Proofs Involving Sets
• Chapter 9: Disproof
• Chapter 10: Mathematical Induction
These chapters deal with useful variations, embellishments and conse-
quences of the proof techniques introduced in Chapters 4 through 6.
PART IV Relations, Functions and Cardinality
• Chapter 11: Relations
• Chapter 12: Functions
• Chapter 13: Cardinality of Sets
Thesefinalchaptersaremainlyconcernedwiththeideaoffunctions, which
arecentraltoallofmathematics. Uponmasteringthismaterialyouwillbe
ready for advanced mathematics courses such as combinatorics, abstract
algebra, theory of computation, analysis and topology.
x Introduction
To the instructor. The book is designed for a three credit course. Here
is a possible timetable for a fourteen-week semester.
Week Monday Wednesday Friday
1 Section1.1 Section1.2 Sections1.3,1.4
2 Sections1.5,1.6,1.7 Section1.8 Sections1.9∗,2.1
3 Section2.2 Sections2.3,2.4 Sections2.5,2.6
4 Section2.7 Sections2.8∗,2.9 Sections2.10,2.11∗,2.12∗
5 Sections3.1,3.2 Section3.3 Sections3.4,3.5∗
6 EXAM Sections4.1,4.2,4.3 Sections4.3,4.4,4.5∗
7 Sections5.1,5.2,5.3∗ Section6.1 Sections6.26.3∗
8 Sections7.1,7.2∗,7.3 Sections8.1,8.2 Section8.3
9 Section8.4 Sections9.1,9.2,9.3∗ Section10.0
10 Sections10.0,10.3∗ Sections10.1,10.2 EXAM
11 Sections11.0,11.1 Sections11.2,11.3 Sections11.4,11.5
12 Section12.1 Section12.2 Section12.2
13 Sections12.3,12.4∗ Section12.5 Sections12.5,12.6∗
14 Section13.1 Section13.2 Sections13.3,13.4∗
Sections marked with ∗ may require only the briefest mention in class, or
may be best left for the students to digest on their own. Some instructors
may prefer to omit Chapter 3.
Acknowledgments. I thank my students in VCU’s MATH 300 courses
for offering feedback as they read the first edition of this book. Thanks
especially to Cory Colbert and Lauren Pace for rooting out typographical
mistakes and inconsistencies. I am especially indebted to Cory for reading
early drafts of each chapter and catching numerous mistakes before I
posted the final draft on my web page. Cory also created the index,
suggested some interesting exercises, and wrote some solutions. Thanks
to Andy Lewis and Sean Cox for suggesting many improvements while
teaching from the book. I am indebted to Lon Mitchell, whose expertise
with typesetting and on-demand publishing made the print version of this
book a reality.
And thanks to countless readers all over the world who contacted me
concerning errors and omissions. Because of you, this is a better book.