Third Edition Mathematical Proofs A Transition to Advanced Mathematics Gary Chartrand

489 Pages·2012·3.45 MB·English

T h i r d E d i t i o n
Mathematical Proofs
A Transition to
Advanced Mathematics
Gary Chartrand
WesternMichiganUniversity
Albert D. Polimeni
StateUniversityofNewYorkatFredonia
Ping Zhang
WesternMichiganUniversity
Boston Columbus Indianapolis NewYork SanFrancisco UpperSaddleRiver
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LibraryofCongressCataloging-in-PublicationData
Chartrand,Gary.
Mathematicalproofs:atransitiontoadvancedmathematics/GaryChartrand,
AlbertD.Polimeni,PingZhang.–3rded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN-13:978-0-321-79709-4
ISBN-10:0-321-79709-4
1.Prooftheory—Textbooks. I.Polimeni,AlbertD.,1938– II.Zhang,Ping,
1957– III.Title.
QA9.54.C48 2013
511.3(cid:3)6—dc23 2012012552
10987654321—CW—1615141312
ISBN-13:978-0-321-79709-4
ISBN-10: 0-321-79709-4
To
the memory of my mother and father G.C.
the memory of my uncle Joe and my brothers John and Rocky A.D.P.
my mother and the memory of my father P.Z.
Contents
0
Communicating Mathematics 1
LearningMathematics 2
WhatOthersHaveSaidAboutWriting 4
MathematicalWriting 5
UsingSymbols 6
WritingMathematicalExpressions 8
CommonWordsandPhrasesinMathematics 10
SomeClosingCommentsAboutWriting 12
1
Sets 14
1.1 DescribingaSet 14
1.2 Subsets 18
1.3 SetOperations 21
1.4 IndexedCollectionsofSets 24
1.5 PartitionsofSets 27
1.6 CartesianProductsofSets 28
ExercisesforChapter1 29
2
Logic 37
2.1 Statements 37
2.2 TheNegationofaStatement 39
2.3 TheDisjunctionandConjunctionofStatements 41
2.4 TheImplication 42
2.5 MoreonImplications 44
2.6 TheBiconditional 47
2.7 TautologiesandContradictions 49
iv
Contents v
2.8 LogicalEquivalence 51
2.9 SomeFundamentalPropertiesofLogicalEquivalence 53
2.10 QuantifiedStatements 55
2.11 CharacterizationsofStatements 63
ExercisesforChapter2 64
3
Direct Proof and Proof by Contrapositive 77
3.1 TrivialandVacuousProofs 78
3.2 DirectProofs 80
3.3 ProofbyContrapositive 84
3.4 ProofbyCases 89
3.5 ProofEvaluations 92
ExercisesforChapter3 93
4
More on Direct Proof and Proof by Contrapositive 99
4.1 ProofsInvolvingDivisibilityofIntegers 99
4.2 ProofsInvolvingCongruenceofIntegers 103
4.3 ProofsInvolvingRealNumbers 105
4.4 ProofsInvolvingSets 108
4.5 FundamentalPropertiesofSetOperations 111
4.6 ProofsInvolvingCartesianProductsofSets 113
ExercisesforChapter4 114
5
Existence and Proof by Contradiction 120
5.1 Counterexamples 120
5.2 ProofbyContradiction 124
5.3 AReviewofThreeProofTechniques 130
5.4 ExistenceProofs 132
5.5 DisprovingExistenceStatements 136
ExercisesforChapter5 137
6
Mathematical Induction 142
6.1 ThePrincipleofMathematicalInduction 142
6.2 AMoreGeneralPrincipleofMathematicalInduction 151
6.3 ProofbyMinimumCounterexample 158
6.4 TheStrongPrincipleofMathematicalInduction 161
ExercisesforChapter6 165
vi Contents
7
Prove or Disprove 170
7.1 ConjecturesinMathematics 170
7.2 RevisitingQuantifiedStatements 173
7.3 TestingStatements 178
ExercisesforChapter7 185
8
Equivalence Relations 192
8.1 Relations 192
8.2 PropertiesofRelations 193
8.3 EquivalenceRelations 196
8.4 PropertiesofEquivalenceClasses 198
8.5 CongruenceModulon 202
8.6 TheIntegersModulon 207
ExercisesforChapter8 210
9
Functions 216
9.1 TheDefinitionofFunction 216
9.2 TheSetofAllFunctionsfrom Ato B 219
9.3 One-to-OneandOntoFunctions 220
9.4 BijectiveFunctions 222
9.5 CompositionofFunctions 225
9.6 InverseFunctions 229
9.7 Permutations 232
ExercisesforChapter9 234
10
Cardinalities of Sets 242
10.1 NumericallyEquivalentSets 243
10.2 DenumerableSets 244
10.3 UncountableSets 250
10.4 ComparingCardinalitiesofSets 255
10.5 TheSchro¨der–BernsteinTheorem 258
ExercisesforChapter10 262
11
Proofs in Number Theory 266
11.1 DivisibilityPropertiesofIntegers 266
11.2 TheDivisionAlgorithm 267
Contents vii
11.3 GreatestCommonDivisors 271
11.4 TheEuclideanAlgorithm 272
11.5 RelativelyPrimeIntegers 275
11.6 TheFundamentalTheoremofArithmetic 277
11.7 ConceptsInvolvingSumsofDivisors 280
ExercisesforChapter11 281
12
Proofs in Calculus 288
12.1 LimitsofSequences 288
12.2 InfiniteSeries 295
12.3 LimitsofFunctions 300
12.4 FundamentalPropertiesofLimitsofFunctions 307
12.5 Continuity 312
12.6 Differentiability 314
ExercisesforChapter12 317
13
Proofs in Group Theory 322
13.1 BinaryOperations 322
13.2 Groups 326
13.3 PermutationGroups 330
13.4 FundamentalPropertiesofGroups 333
13.5 Subgroups 336
13.6 IsomorphicGroups 340
ExercisesforChapter13 344
14
Proofs in Ring Theory (Online)
14.1 Rings
14.2 ElementaryPropertiesofRings
14.3 Subrings
14.4 IntegralDomains
14.5 Fields
ExercisesforChapter14
15
Proofs in Linear Algebra (Online)
15.1 PropertiesofVectorsin3-Space
15.2 VectorSpaces
15.3 Matrices
15.4 SomePropertiesofVectorSpaces
viii Contents
15.5 Subspaces
15.6 SpansofVectors
15.7 LinearDependenceandIndependence
15.8 LinearTransformations
15.9 PropertiesofLinearTransformations
ExercisesforChapter15
16
Proofs in Topology (Online)
16.1 MetricSpaces
16.2 OpenSetsinMetricSpaces
16.3 ContinuityinMetricSpaces
16.4 TopologicalSpaces
16.5 ContinuityinTopologicalSpaces
ExercisesforChapter16
Answers and Hints to Selected Odd-Numbered
Exercises in Chapters 14–16 (online)
Answers and Hints to Odd-Numbered Section Exercises 351
References 394
Index of Symbols 395
Index 396
PREFACE TO THE THIRD
EDITION
Aswementionedintheprefacesofthefirsttwoeditions,becausetheteachingofcalculus
in many colleges and universities has become more problem-oriented with added em-
phasisontheuseofcalculatorsandcomputers,thetheoreticalgapbetweenthematerial
presentedincalculusandthemathematicalbackgroundexpected(oratleasthopedfor)
inmoreadvancedcoursessuchasabstractalgebraandadvancedcalculushaswidened.
In an attempt to narrow this gap and to better prepare students for the more abstract
mathematicscoursestofollow,manycollegesanduniversitieshaveintroducedcourses
that are now commonly called transition courses. In these courses, students are intro-
ducedtoproblemswhosesolutioninvolvesmathematicalreasoningandaknowledgeof
prooftechniquesandwritingclearproofs.Topicssuchasrelations,functionsandcardi-
nalitiesofsetsareencounteredthroughouttheoreticalmathematicscourses.Inaddition,
transition courses often include theoretical aspects of number theory, abstract algebra,
andcalculus.Thistextbookhasbeenwrittenforsuchacourse.
The idea for this textbook originated in the early 1980s, long before transition
courses became fashionable, during the supervising of undergraduate mathematics re-
search projects. We came to realize that even advanced undergraduates lack a sound
understanding of proof techniques and have difficulty writing correct and clear proofs.
Atthattime,asetofnoteswasdevelopedforthesestudents.Thiswasfollowedbythe
introduction of a transition course, for which a more detailed set of notes was written.
Thefirsteditionofthisbookemanatedfromthesenotes,whichinturnhasledtoasecond
editionandnowthisthirdedition.
Whileunderstandingproofsandprooftechniquesandwritinggoodproofsaremajor
goalshere,thesearenotthingsthatcanbeaccomplishedtoanygreatdegreeinasingle
courseduringasinglesemester.Thesemustcontinuetobeemphasizedandpracticedin
succeedingmathematicscourses.
Our Approach
Sincethistextbookoriginatedfromnotesthatwerewrittenexclusivelyforundergradu-
atestohelpthemunderstandprooftechniquesandtowritegoodproofs,thisisthetone
ix