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P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO A01_CHART6753_04_SE_FM PH03348-Chartrand September22,2017 8:50 CharCount=0 Fourth Edition Mathematical Proofs A Transition to Advanced Mathematics Gary Chartrand WesternMichiganUniversity Albert D. Polimeni StateUniversityofNewYorkatFredonia Ping Zhang WesternMichiganUniversity i Copyright(cid:2)C 2018,2013,2008byPearsonEducation,Inc. LibraryofCongressCataloging-in-PublicationData Names:Chartrand,Gary,author.|Polimeni,AlbertD.,1938-author.|Zhang,Ping,1957-author. Title:Mathematicalproofs:atransitiontoadvancedmathematics/GaryChartrand, WesternMichiganUniversity,AlbertD.Polimeni,StateUniversityofNewYorkatFredonia, PingZhang,WesternMichiganUniversity. Description:Fourthedition.|Boston:Pearson,[2018]|Includesbibliographicalreferences andindex. Identifiers:LCCN2017024934|ISBN9780134746753|ISBN0134746759 Subjects:LCSH:Prooftheory–Textbooks. Classification:LCCQA9.54.C482018|DDC511.3/6–dc23LCrecordavailableat https://lccn.loc.gov/2017024934 ISBN-13:978-0-13-474675-3 ISBN-10: 0-13-474675-9 ii P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO A01_CHART6753_04_SE_FM PH03348-Chartrand September22,2017 8:50 CharCount=0 Contents 0 Communicating Mathematics 1 0.1 LearningMathematics 2 0.2 WhatOthersHaveSaidAboutWriting 3 0.3 MathematicalWriting 5 0.4 UsingSymbols 6 0.5 WritingMathematicalExpressions 8 0.6 CommonWordsandPhrasesinMathematics 10 0.7 SomeClosingCommentsAboutWriting 12 1 Sets 14 1.1 DescribingaSet 14 1.2 Subsets 18 1.3 SetOperations 23 1.4 IndexedCollectionsofSets 27 1.5 PartitionsofSets 31 1.6 CartesianProductsofSets 33 Chapter1SupplementalExercises 35 2 Logic 38 2.1 Statements 38 2.2 Negations 41 2.3 DisjunctionsandConjunctions 43 2.4 Implications 45 2.5 MoreonImplications 49 2.6 Biconditionals 53 2.7 TautologiesandContradictions 57 2.8 LogicalEquivalence 60 2.9 SomeFundamentalPropertiesofLogicalEquivalence 62 2.10 QuantifiedStatements 65 2.11 Characterizations 76 Chapter2SupplementalExercises 78 3 Direct Proof and Proof by Contrapositive 81 3.1 TrivialandVacuousProofs 82 3.2 DirectProofs 85 3.3 ProofbyContrapositive 89 3.4 ProofbyCases 94 3.5 ProofEvaluations 98 Chapter3SupplementalExercises 102 4 More on Direct Proof and Proof by Contrapositive 105 4.1 ProofsInvolvingDivisibilityofIntegers 105 4.2 ProofsInvolvingCongruenceofIntegers 110 4.3 ProofsInvolvingRealNumbers 113 4.4 ProofsInvolvingSets 117 4.5 FundamentalPropertiesofSetOperations 120 4.6 ProofsInvolvingCartesianProductsofSets 122 Chapter4SupplementalExercises 123 5 Existence and Proof by Contradiction 127 5.1 Counterexamples 127 5.2 ProofbyContradiction 131 5.3 AReviewofThreeProofTechniques 138 5.4 ExistenceProofs 141 5.5 DisprovingExistenceStatements 146 Chapter5SupplementalExercises 149 6 Mathematical Induction 152 6.1 ThePrincipleofMathematicalInduction 152 6.2 AMoreGeneralPrincipleofMathematicalInduction 162 6.3 TheStrongPrincipleofMathematicalInduction 170 6.4 ProofbyMinimumCounterexample 174 Chapter6SupplementalExercises 178 7 Reviewing Proof Techniques 181 7.1 ReviewingDirectProofandProofbyContrapositive 182 7.2 ReviewingProofbyContradictionandExistenceProofs 185 7.3 ReviewingInductionProofs 188 7.4 ReviewingEvaluationsofProposedProofs 189 ExercisesforChapter7 193 8 Prove or Disprove 200 8.1 ConjecturesinMathematics 200 8.2 RevisitingQuantifiedStatements 205 8.3 TestingStatements 211 Chapter8SupplementalExercises 220 9 Equivalence Relations 224 9.1 Relations 224 9.2 PropertiesofRelations 226 9.3 EquivalenceRelations 230 9.4 PropertiesofEquivalenceClasses 235 9.5 CongruenceModulon 239 9.6 TheIntegersModulon 245 Chapter9SupplementalExercises 248 10 Functions 251 10.1 TheDefinitionofFunction 251 10.2 One-to-oneandOntoFunctions 256 10.3 BijectiveFunctions 259 10.4 CompositionofFunctions 263 10.5 InverseFunctions 267 Chapter10SupplementalExercises 274 11 Cardinalities of Sets 278 11.1 NumericallyEquivalentSets 279 11.2 DenumerableSets 280 11.3 UncountableSets 288 11.4 ComparingCardinalitiesofSets 293 11.5 TheSchro¨der-BernsteinTheorem 296 Chapter11SupplementalExercises 301 12 Proofs in Number Theory 303 12.1 DivisibilityPropertiesofIntegers 303 12.2 TheDivisionAlgorithm 305 12.3 GreatestCommonDivisors 310 12.4 TheEuclideanAlgorithm 312 12.5 RelativelyPrimeIntegers 315 12.6 TheFundamentalTheoremofArithmetic 318 12.7 ConceptsInvolvingSumsofDivisors 322 Chapter12SupplementalExercises 324 13 Proofs in Combinatorics 327 13.1 TheMultiplicationandAdditionPrinciples 327 13.2 ThePrincipleofInclusion-Exclusion 333 13.3 ThePigeonholePrinciple 336 13.4 PermutationsandCombinations 340 13.5 ThePascalTriangle 348 13.6 TheBinomialTheorem 352 13.7 PermutationsandCombinationswithRepetition 357 Chapter13SupplementalExercises 363 14 Proofs in Calculus 365 14.1 LimitsofSequences 365 14.2 InfiniteSeries 373 14.3 LimitsofFunctions 378 14.4 FundamentalPropertiesofLimitsofFunctions 386 14.5 Continuity 392 14.6 Differentiability 395 Chapter14SupplementalExercises 397 15 Proofs in Group Theory 400 15.1 BinaryOperations 400 15.2 Groups 405 15.3 PermutationGroups 411 15.4 FundamentalPropertiesofGroups 414 15.5 Subgroups 418 15.6 IsomorphicGroups 423 Chapter15SupplementalExercises 428 16 Proofs in Ring Theory (Onlineatgoo.gl/zKdQor) 16.1 Rings 16.2 ElementaryPropertiesofRings 16.3 Subrings 16.4 IntegralDomains 16.5 Fields ExercisesforChapter16 17 Proofs in Linear Algebra (Onlineatgoo.gl/Tmh2ZB) 17.1 PropertiesofVectorsin3-Space 17.2 VectorSpaces 17.3 Matrices 17.4 SomePropertiesofVectorSpaces 17.5 Subspaces 17.6 SpansofVectors 17.7 LinearDependenceandIndependence 17.8 LinearTransformations 17.9 PropertiesofLinearTransformations ExercisesforChapter17 18 Proofs with Real and Complex Numbers (Onlineatgoo.gl/ymv5kJ) 18.1 TheRealNumbersasanOrderedField 18.2 TheRealNumbersandtheCompletenessAxiom 18.3 OpenandClosedSetsofRealNumbers 18.4 CompactSetsofRealNumbers 18.5 ComplexNumbers 18.6 DeMoivre’sTheoremandEuler’sFormula ExercisesforChapter18 19 Proofs in Topology (Onlineatgoo.gl/bWFRMu) 19.1 MetricSpaces 19.2 OpenSetsinMetricSpaces 19.3 ContinuityinMetricSpaces 19.4 TopologicalSpaces 19.5 ContinuityinTopologicalSpaces ExercisesforChapter19 AnswersandHintstoSelectedOdd-Numbered ExercisesinChapters16–19(onlineatgoo.gl/uPLFfs) Answers to Odd-Numbered Section Exercises 430 References 483 Credits 484 Index of Symbols 486 Index 487 P1:OSO/OVY P2:OSO/OVY QC:OSO/OVY T1:OSO A01_CHART6753_04_SE_FM PH03348-Chartrand September22,2017 8:50 CharCount=0 Preface to the Fourth Edition Astheteachingofcalculusinmanycollegesanduniversitieshasbecomemoreproblem- oriented with added emphasis on the use of calculators and computers, the theoretical gap between the material presented in calculus and the mathematical background expected(oratleasthopedfor)inadvancedcalculusandothermoreadvancedcourses has widened. In an attempt to narrow this gap and to better prepare students for the more abstract mathematics courses to follow, many colleges and universities have introducedcoursesthatarenowcommonlycalled“transitioncourses.”Inthesecourses, students are introduced to problems whose solution involves mathematical reasoning andaknowledgeofprooftechniques,wherewritingclearproofsisemphasized.Topics such as relations, functions and cardinalities of sets are encountered throughout theo- reticalmathematicscourses.Lastly,transitioncoursesoftenincludetheoreticalaspects ofnumbertheory,combinatorics,abstractalgebraandcalculus.Thistextbookhasbeen writtenforsuchacourse. The idea for this textbook originated in the early 1980s, long before transition courses became fashionable, during the supervision of undergraduate mathematics re- search projects. We came to realize that even advanced undergraduates lack a sound understandingofprooftechniquesandhavedifficultywritingcorrectandclearproofs. Atthattime,wedevelopedasetofnotesforthesestudents.Thiswasfollowedbythe introductionofatransitioncourse,forwhichamoredetailedsetofnoteswaswritten. The firstedition of this book emanated fromthese notes, which in turn has ultimately ledtothisfourthedition. Whileunderstandingproofsandprooftechniquesandwritinggoodproofsaremajor goalshere,thesearenotthingsthatcanbeaccomplishedtoanygreatdegreeinasingle courseduringasinglesemester.Thesemustcontinuetobeemphasizedandpracticedin succeedingmathematicscourses. Our Approach Sincethistextbookoriginatedfromnotesthatwerewrittenexclusivelyforundergrad- uates to help them understand proof techniques and to write good proofs, the tone is student-friendly.Numerousexamplesofproofsarepresentedinthetext.Followingcom- monpractice,weindicatetheendofaproofwiththesquaresymbol .Oftenweprecede a proof by a discussion, referred to as a proof strategy, where we think through what is needed to present a proof of the result in question. Other times, we find it useful to reflectonaproofwehavejustpresentedtopointoutcertainkeydetails.Werefertoa discussionofthistypeasaproofanalysis.Periodically,wepresentandsolveproblems, andwemayfinditconvenienttodiscusssomefeaturesofthesolution,whichwereferto simplyasananalysis.Forclarity,weindicatetheendofadiscussionofaproofstrategy, proofanalysis,analysis,orsolutionofanexamplewiththediamondsymbol(cid:2). Amajorgoalofthistextbookistohelpstudentslearntoconstructproofsoftheir ownthatarenotonlymathematicallycorrectbutclearlywritten.Moreadvancedmathe- maticsstudentsshouldstrivetopresentproofsthatareconvincing,readable,notationally consistent and grammatically correct. A secondary goal is to have students gain suffi- cientknowledgeofandconfidencewithproofssothattheywillrecognize,understand andappreciateaproofthatisproperlywritten. Aswiththefirstthreeeditions,thefourtheditionofthisbookisintendedtoassist thestudentinmakingthetransitiontocoursesthatrelymoreonmathematicalproofand reasoning.Weenvisionthatstudentstakingacoursebasedonthisbookhaveprobably hadayearofcalculus(andpossiblyanothercoursesuchaselementarylinearalgebra) althoughnospecificprerequisitemathematicscoursesarerequired.Itislikelythat,prior to taking this course, a student’s training in mathematics consisted primarily of doing patterned problems; that is, students have been taught methods for solving problems, likelyincludingsomeexplanationastowhythesemethodsworked.Studentsmayvery well have had exposure to some proofs in earlier courses but, more than likely, were unawareofthelogicinvolvedandofthemethodofproofbeingused.Theremayhave evenbeentimeswhenthestudentswerenotcertainwhatwasbeingproved. New to This Edition The following changes and additions to the third edition have resulted in this fourth editionofthetext: • PresentationslidesinPDFandLaTeXformatshavebeencreatedtoaccompany everychapter.Thesepresentationsprovideexamplesandexpositiononkeytopics. UsingshortURLs(whicharecalledoutinthemarginusingthe icon),these presentationsarelinkedinthetextbesidethesupplementalexercisesattheendof eachchapter.Theseslidescanbeusedbyinstructorsinlecture,orbystudentsto learnandreviewkeyideas. • ThenewChapter7,“ReviewingProofTechniques,”summarizesallthetechniques thathavebeenpresented.Theplacementofthischapterallowsinstructorsand studentstoreviewallofthesetechniquesbeforebeginningtoexploredifferent contextsinwhichproofsareused.Thisnewchapterincludesmanynewexamples andexercises. • ThenewChapter13,“ProofsinCombinatorics,”hasbeenaddedbecauseofa demandforsuchachapter.Here,resultsandexamplesarepresentedinthis importantareaofdiscretemathematics.Numerousexercisesarealsoincludedin thischapter. • ThenewonlineChapter18,“ProofswithRealandComplexNumbers,”hasbeen addedtoprovideinformationonthesetwoimportantclassesofnumbers.This chapterincludesmanyimportantclassicalresultsonrealnumbersaswellas importantresultsfromcomplexvariables.Ineachcase,detailedproofsaregiven. Chapters16(ProofsinRingTheory),17(ProofsinLinearAlgebra),and19(Proofs inTopology)continuetoexistonlinetoallowfacultytotailorthecoursetomeet theirspecificneeds. • Morethan250exerciseshavebeenadded.Manyofthenewexercisesfallintothe moderatedifficultylevelandrequiremorethoughttosolve. • Sectionexercisesforeachchapterhavebeenmovedfromtheendofthechapterto theendofeachsectionwithinachapter.(Exceptions:forthereviewchapter[7]and theonlinechapters[16–19],allexercisesremainattheendofthechapter.) • Inthepreviousedition,therewereexercisesattheendofeachchaptercalled AdditionalExercises.Thesesummativeexercisesservedtopulltogethertheideas fromthevarioussectionsofthechapter.Theseexercisesremainattheendofthe chaptersinthisedition,buttheyhavebeenrenamedSupplementalExercises. Contents and Structure Outline of Contents EachoftheChapters1–6and8–15isdividedintosections,andexercisesforeachsection occurattheendofthatsection.Thereisalsoafinalsupplementalsectionofexercises fortheentirechapterappearingattheendofthatchapter. Since writinggood proofs requires a certaindegree of competence inwriting,we havedevotedChapter0towritingmathematics.Theemphasisofthischapterisoneffec- tiveandclearexposition,correctusageofsymbols,writinganddisplayingmathematical expressions, and using key words and phrases. Although every instructor will empha- sizewritinginhisorherownway,wefeelthatitisusefultoreadChapter0periodically throughoutthecourse.Itwillmeanmoreasthestudentprogressesthroughthecourse. Chapter 1 contains a gentle introduction to sets, so that everyone has the same backgroundandisusingthesamenotationasweprepareforwhatliesahead.Noproofs involvingsetsoccuruntilChapter4.MuchofChapter1mayverywellbeareviewfor many. Chapter2dealsexclusivelywithlogic.Thegoalhereistopresentwhatisneeded to get into proofs as quickly as possible. Much of the emphasis in Chapter 2 is on statements,implicationsandquantifiedstatements,includingadiscussionofmixedquan- tifiers.Setsareintroducedbeforelogicsothatthestudent’sfirstencounterwithmathe- maticshereisafamiliaroneandbecausesetsareneededtodiscussquantifiedstatements properlyinChapter2. Thetwoprooftechniquesofdirectproofandproofbycontrapositiveareintroduced inChapter3inthefamiliarsettingofevenandoddintegers.Proofbycasesisdiscussed inthischapteraswellasproofsof“ifandonlyif”statements.Chapter4continuesthis discussion in other settings, namely divisibility of integers, congruence, real numbers andsets. ThetechniqueofproofbycontradictionisintroducedinChapter5.Sinceexistence proofsandcounterexampleshaveaconnectionwithproofbycontradiction,thesealso occurinChapter5.Thetopicofuniqueness(ofanelementwithspecifiedproperties)is alsoaddressedinChapter5.

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Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.