Book of Proof

20130101·1.4 MB·English

by  Prof. Richard Hammack

Book of Proof
Richard Hammack
Virginia Commonwealth University
RichardHammack(publisher)
DepartmentofMathematics&AppliedMathematics
P.O.Box842014
VirginiaCommonwealthUniversity
Richmond,Virginia,23284
BookofProof
Edition2
©2013byRichardHammack
This work is licensed under the Creative Commons Attribution-No Derivative Works 3.0
License
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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 28
1.10. Russell’sParadox 29
2. Logic 31
2.1. Statements 32
2.2. And,Or,Not 36
2.3. ConditionalStatements 39
2.4. BiconditionalStatements 42
2.5. TruthTablesforStatements 44
2.6. LogicalEquivalence 47
2.7. Quantifiers 49
2.8. MoreonConditionalStatements 52
2.9. TranslatingEnglishtoSymbolicLogic 53
2.10. NegatingStatements 55
2.11. LogicalInference 59
2.12. AnImportantNote 60
3. Counting 61
3.1. CountingLists 61
3.2. Factorials 68
3.3. CountingSubsets 71
3.4. Pascal’sTriangleandtheBinomialTheorem 76
3.5. Inclusion-Exclusion 79
v
II How to Prove Conditional Statements
4. Direct Proof 85
4.1. Theorems 85
4.2. Definitions 87
4.3. DirectProof 90
4.4. UsingCases 96
4.5. TreatingSimilarCases 97
5. Contrapositive Proof 100
5.1. ContrapositiveProof 100
5.2. CongruenceofIntegers 103
5.3. MathematicalWriting 105
6. Proof by Contradiction 109
6.1. ProvingStatementswithContradiction 110
6.2. ProvingConditionalStatementsbyContradiction 113
6.3. CombiningTechniques 114
6.4. SomeWordsofAdvice 115
III More on Proof
7. Proving Non-Conditional Statements 119
7.1. If-and-Only-IfProof 119
7.2. EquivalentStatements 121
7.3. ExistenceProofs;ExistenceandUniquenessProofs 122
7.4. ConstructiveVersusNon-ConstructiveProofs 126
8. Proofs Involving Sets 129
8.1. HowtoProve a∈A 129
8.2. HowtoProve A⊆B 131
8.3. HowtoProve A=B 134
8.4. Examples: PerfectNumbers 137
9. Disproof 144
9.1. Counterexamples 146
9.2. DisprovingExistenceStatements 148
9.3. DisproofbyContradiction 150
10. Mathematical Induction 152
10.1. ProofbyStrongInduction 159
10.2. ProofbySmallestCounterexample 163
10.3. FibonacciNumbers 165
vi
IV Relations, Functions and Cardinality
11. Relations 173
11.1. PropertiesofRelations 177
11.2. EquivalenceRelations 182
11.3. EquivalenceClassesandPartitions 186
11.4. TheIntegersModulo n 189
11.5. RelationsBetweenSets 192
12. Functions 194
12.1. Functions 194
12.2. InjectiveandSurjectiveFunctions 199
12.3. ThePigeonholePrinciple 203
12.4. Composition 206
12.5. InverseFunctions 209
12.6. ImageandPreimage 212
13. Cardinality of Sets 215
13.1. SetswithEqualCardinalities 215
13.2. CountableandUncountableSets 221
13.3. ComparingCardinalities 226
13.4. TheCantor-Bernstein-SchröederTheorem 230
Conclusion 237
Solutions 238
Index 299
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
versions (produced through an on-demand press) cost 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 twelve years. It is written for
an audience of mathematics majors at Virginia Commonwealth University,
a large state university. It caters to our program, and is intended to
prepare our students for our more advanced courses. However, 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 approach to mathematics that is more theoretical
than computational. In this approach, 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
are central to all of mathematics. Upon mastering this material you will
be ready for many advanced mathematics courses, such as combinatorics,
abstract algebra, analysis and topology.
x Introduction
To the instructor. The book is designed to be covered in a fourteen-week
semester. Here is a possible timetable.
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.
Finally, thanks to countless readers all over the world who contacted
me concerning errors and omissions. Because of you, this is a better book.