A transition to abstract mathematics: mathematical thinking and writing

379 Pages·2008·1.88 MB·English

by  Randall Maddox

A Transition to Abstract Mathematics
Mathematical Thinking and Writing
Second Edition
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A Transition to Abstract
Mathematics
Mathematical Thinking and Writing
Second Edition
Randall B. Maddox
Pepperdine University
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Maddox,RandallB.
Atransitiontoabstractmathematics:learningmathematicalthinkingand
writing/RandallB.Maddox.–2nded.
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ISBN978-0-12-374480-7(hardcover:acid-freepaper)1.Prooftheory.2.Logic,Symbolic
andmathematical.I.Maddox,RandallB.Mathematicalthinkingandwriting.II.Title.
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Contents
Why Read This Book? xiii
Preface xv
Preface to the First Edition xvii
Acknowledgments xxi
0 Notation and Assumptions 1
0.1 SetTerminologyandNotation 1
0.2 AssumptionsabouttheRealNumbers 3
0.2.1 BasicAlgebraicProperties 3
0.2.2 OrderingProperties 5
0.2.3 OtherAssumptions 7
I Foundations of Logic and Proof Writing 9
1 Language and Mathematics 11
1.1 IntroductiontoLogic 11
1.1.1 Statements 11
1.1.2 NegationofaStatement 13
1.1.3 CombiningStatementswithAND 13
1.1.4 CombiningStatementswithOR 14
1.1.5 LogicalEquivalence 16
1.1.6 TautologiesandContradictions 18
vii
viii Contents
1.2 If-ThenStatements 18
1.2.1 If-ThenStatementsDefined 18
1.2.2 Variationsonp→q 21
1.2.3 LogicalEquivalenceandTautologies 23
1.3 UniversalandExistentialQuantifiers 27
1.3.1 TheUniversalQuantifier 28
1.3.2 TheExistentialQuantifier 29
1.3.3 UniqueExistence 32
1.4 NegationsofStatements 33
1.4.1 NegationsofANDandORStatements 33
1.4.2 NegationsofIf-ThenStatements 34
1.4.3 NegationsofStatementswiththeUniversalQuantifier 36
1.4.4 NegationsofStatementswiththeExistentialQuantifier 37
1.5 HowWeWriteProofs 40
1.5.1 DirectProof 40
1.5.2 ProofbyContrapositive 41
1.5.3 ProvingaLogicallyEquivalentStatement 41
1.5.4 ProofbyContradiction 42
1.5.5 DisprovingaStatement 42
2 Properties of Real Numbers 45
2.1 BasicAlgebraicPropertiesofRealNumbers 45
2.1.1 PropertiesofAddition 46
2.1.2 PropertiesofMultiplication 49
2.2 OrderingPropertiesoftheRealNumbers 51
2.3 AbsoluteValue 53
2.4 TheDivisionAlgorithm 56
2.5 DivisibilityandPrimeNumbers 59
3 Sets and Their Properties 63
3.1 SetTerminology 63
3.2 ProvingBasicSetProperties 67
3.3 FamiliesofSets 71
3.4 ThePrincipleofMathematicalInduction 78
3.5 VariationsofthePMI 85
3.6 EquivalenceRelations 91
3.7 EquivalenceClassesandPartitions 97
3.8 BuildingtheRationalNumbers 102
3.8.1 DefiningRationalEquality 103
3.8.2 RationalAdditionandMultiplication 104
3.9 RootsofRealNumbers 106
Contents ix
3.10 IrrationalNumbers 107
3.11 RelationsinGeneral 111
4 Functions 119
4.1 DefinitionandExamples 119
4.2 One-to-oneandOntoFunctions 125
4.3 ImageandPre-ImageSets 128
4.4 CompositionandInverseFunctions 131
4.4.1 CompositionofFunctions 132
4.4.2 InverseFunctions 133
4.5 ThreeHelpfulTheorems 135
4.6 FiniteSets 137
4.7 InfiniteSets 139
4.8 CartesianProductsandCardinality 144
4.8.1 CartesianProducts 144
4.8.2 FunctionsBetweenFiniteSets 146
4.8.3 Applications 148
4.9 CombinationsandPartitions 151
4.9.1 Combinations 151
4.9.2 PartitioningaSet 152
4.9.3 Applications 153
4.10 TheBinomialTheorem 157
II Basic Principles of Analysis 163
5 The Real Numbers 165
5.1 TheLeastUpperBoundAxiom 165
5.1.1 LeastUpperBounds 166
5.1.2 GreatestLowerBounds 168
5.2 TheArchimedeanProperty 169
5.2.1 MaximumandMinimumofFiniteSets 170
5.3 OpenandClosedSets 172
5.4 Interior,Exterior,Boundary,andClusterPoints 175
5.4.1 Interior,Exterior,andBoundary 175
5.4.2 ClusterPoints 176
5.5 ClosureofSets 178
5.6 Compactness 180
6 Sequences of Real Numbers 185
6.1 SequencesDefined 185
6.1.1 MonotoneSequences 186
6.1.2 BoundedSequences 187