20130101·1.4 MB·English

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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 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 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.

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