Mathematical Thinking and Writing : A Transition to Abstract Mathematics

323 Pages·2007·3.3 MB·English

by  Maddox, Randall B.

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MATHEMATICAL
T H I N K I N G
A N D W R I T I N G
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RANDALL B. MADDOX
Pepperdine University, Malibu, CA
MATHEMATICAL
T H I N K I N G
A N D W R I T I N G
A Transition to Abstract Mathematics
A C A D E M I C P R E S S
A Harcourt Science and Technology Company
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Contents
WhyReadThisBook xiii
Preface xv
CHAPTER0 NotationandAssumptions 1
0.1 SetTerminologyandNotation 1
0.2 Assumptions 5
0.2.1 Basicalgebraicpropertiesofrealnumbers 5
0.2.2 Orderingofrealnumbers 7
0.2.3 OtherassumptionsaboutR 9
PARTI FOUNDATIONS OF LOGIC AND PROOF WRITING 11
CHAPTER1 Logic 13
1.1 IntroductiontoLogic 13
1.1.1 Statements 13
1.1.2 Negationofastatement 15
1.1.3 CombiningstatementswithAND/OR 15
1.1.4 Logicalequivalence 18
1.1.5 Tautologiesandcontradictions 18
1.2 If-ThenStatements 20
1.2.1 If-thenstatements 20
1.2.2 Variationsonp→q 22
1.2.3 Logicalequivalenceandtautologies 23
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viii Contents
1.3 UniversalandExistentialQuantifiers 27
1.3.1 Theuniversalquantifier 27
1.3.2 Theexistentialquantifier 29
1.3.3 Uniqueexistence 30
1.4 NegationsofStatements 31
1.4.1 Negationsofp∧qandp∨q 32
1.4.2 Negationsofp→q 33
1.4.3 Negationsofstatements
with∀and∃ 33
CHAPTER2 Beginner-LevelProofs 38
2.1 ProofsInvolvingSets 38
2.1.1 Termsinvolvingsets 38
2.1.2 Directproofs 41
2.1.3 Proofsbycontrapositive 44
2.1.4 Proofsbycontradiction 45
2.1.5 Disprovingastatement 45
2.2 IndexedFamiliesofSets 47
2.3 AlgebraicandOrderingPropertiesofR 53
2.3.1 Basicalgebraicproperties
ofrealnumbers 53
2.3.2 Orderingoftherealnumbers 56
2.3.3 Absolutevalue 57
2.4 ThePrincipleofMathematicalInduction 61
2.4.1 ThestandardPMI 62
2.4.2 VariationofthePMI 64
2.4.3 Stronginduction 65
2.5 EquivalenceRelations:TheIdeaofEquality 68
2.5.1 Analyzingequality 68
2.5.2 Equivalenceclasses 72
2.6 Equality,Addition,andMultiplicationinQ 76
2.6.1 EqualityinQ 77
2.6.2 Well-defined+and×onQ 78
2.7 TheDivisionAlgorithmandDivisibility 79
2.7.1 Evenandoddintegers;thedivision
algorithm 79
2.7.2 DivisibilityinZ 81
2.8 Rootsandirrationalnumbers 85
2.8.1 Rootsofrealnumbers 86
2.8.2 Existenceofirrationalnumbers 87
2.9 RelationsInGeneral 90
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Contents ix
CHAPTER3 Functions 97
3.1 DefinitionsandTerminology 97
3.1.1 Definitionandexamples 97
3.1.2 Otherterminologyandnotation 101
3.1.3 Threeimportanttheorems 103
3.2 CompositionandInverseFunctions 106
3.2.1 Compositionoffunctions 106
3.2.2 Inversefunctions 108
3.3 CardinalityofSets 110
3.3.1 Finitesets 111
3.3.2 Infinitesets 113
3.4 CountingMethodsandtheBinomialTheorem 118
3.4.1 Theproductrule 118
3.4.2 Permutations 122
3.4.3 Combinationsandpartitions 122
3.4.4 Countingexamples 125
3.4.5 Thebinomialtheorem 126
PARTII BASIC PRINCIPLES OF ANALYSIS 131
CHAPTER4 TheRealNumbers 133
4.1 TheLeastUpperBoundAxiom 134
4.1.1 Leastupperbounds 134
4.1.2 TheArchimedeanpropertyofR 136
4.1.3 Greatestlowerbounds 137
4.1.4 TheLUBandGLBpropertiesapplied
tofinitesets 137
4.2 SetsinR 140
4.2.1 Openandclosedsets 140
4.2.2 Interior,exterior,andboundary 142
4.3 LimitPointsandClosureofSets 143
4.3.1 Closureofsets 144
4.4 Compactness 146
4.5 SequencesinR 149
4.5.1 Monotonesequences 150
4.5.2 Boundedsequences 151
4.6 ConvergenceofSequences 153
4.6.1 Convergencetoarealnumber 154
4.6.2 Convergenceto±∞ 158
4.7 TheNestedIntervalProperty 160
4.7.1 FromLUBaxiomtoNIP 161