Table Of ContentP1: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.
