Passage to Abstract Mathematics

253 Pages·2011·1.54 MB·English

by  Mark E. Watkins

Passage to
Abstract Mathematics
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Passage to
Abstract Mathematics
Mark E. Watkins
Jeffrey L. Meyer
Syracuse University
Addison-Wesley
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LibraryofCongressCataloging-in-PublicationData
Watkins,MarkE.,
Passagetoabstractmathematics/MarkE.WatkinsandJeffreyL.Meyer.
p.cm.
Includesindex.
ISBN978-0-321-73863-9
1.Mathematics—Authorship. 2.Technicalwriting. 3.Logic,Symbolicand
mathematical. I.Meyer,JeffreyL. II.Title.
QA42.W382012
510—dc22
2010029065
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ISBN10: 0-321-73863-2
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Dedicated to our families, our teachers, and our students.
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Contents
Preface for the Instructor xi
Preface for the Student xvii
1 Logic and Proof 1
1.1 Proofs,WhatandWhy? 1
1.2 StatementsandNon-statements 7
1.3 LogicalOperationsandLogicalEquivalence 10
1.4 Conditionals,Tautologies,andContradictions 16
1.5 MethodsofProof 22
1.6 Quantifiers 28
1.7 FurtherExercises 36
2 Numbers 43
2.1 BasicIdeasofSets 43
2.2 SetsofNumbers 44
2.3 SomePropertiesofNandZ 47
2.4 PrimeNumbers 51
2.5 gcd’sandlcm’s 52
2.6 Euclid’sAlgorithm 57
2.7 RationalNumbersandAlgebraicNumbers 60
2.8 FurtherExercises 64
3 Sets 68
3.1 Subsets 68
3.2 OperationswithSets 73
3.3 TheComplementofaSet 74
3.4 TheCartesianProduct 77
vii
viii Contents
3.5 FamiliesofSets 80
3.6 FurtherExercises 83
4 Induction 88
4.1 AnInductiveExample 88
4.2 ThePrincipleofMathematicalInduction 90
4.3 ThePrincipleofStrongInduction 98
4.4 TheBinomialTheorem 103
4.5 FurtherExercises 105
5 Functions 111
5.1 FunctionalNotation 111
5.2 OperationswithFunctions 114
5.3 InducedSetFunctions 120
5.4 Surjections,Injections,andBijections 126
5.5 IdentityFunctions,Cancellation,InverseFunctions,andRestrictions 130
5.6 FurtherExercises 137
6 Binary Relations 142
6.1 Partitions 142
6.2 EquivalenceRelations 144
6.3 OrderRelations 149
6.4 BoundsandExtremalElements 153
6.5 ApplicationstoCalculus 158
6.6 FunctionsRevisited 160
6.7 FurtherExercises 162
7 Infinite Sets and Cardinality 167
7.1 Counting 167
7.2 PropertiesofCountableSets 170
7.3 CountingCountableSets 173
7.4 BinaryRelationsonCardinalNumbers 178
7.5 UncountableSets 181
7.6 FurtherExercises 187
Contents ix
8 Algebraic Systems 190
8.1 BinaryOperations 190
8.2 ModularArithmetic 200
8.3 NumbersRevisited 208
8.4 ComplexNumbers 211
8.5 FurtherExercises 220
Index of Symbols and Notation 225
Index 229