Principles of Mathematics: A Primer

671 Pages·2016·2.66 MB·English

by  Vladimir Lepetic

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PRINCIPLES
OF MATHEMATICS
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PRINCIPLES
OF MATHEMATICS
A Primer
VLADIMIRLEPETIC
DePaulUniversity
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LibraryofCongressCataloging-in-PublicationData:
Lepetic,Vladimir,1950-author.
Principlesofmathematics:aprimer/VladimirLepetic.
pagescm
Includesindex.
ISBN978-1-119-13164-9(cloth)
1.Mathematics–Philosophy.I.Title.
QA8.4.L4472016
510.1–dc23
2015025151
Typesetin10.5/12.5pt,TimesbySPiGlobal,Chennai,India.
PrintedintheUnitedStatesofAmerica
10987654321
1 2016
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ToIvanandMarija
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CONTENTS
Preface xi
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1 SetTheory 1
1.1 Introduction, 1
1.2 SetTheory – Definitions,Notation,andTerminology
– WhatisaSet?, 3
1.3 SetsGivenbyaDefiningProperty, 15
1.4 TheAlgebraofSets, 25
1.5 ThePowerSet, 41
1.6 TheCartesianProduct, 44
1.7 TheSetsN,Z,andQ, 46
1.8 TheSetR – RealNumbersI, 71
1.9 AShortMusingonTransfiniteArithmetic, 80
1.10 TheSetR – RealNumbersII, 102
1.11 SupplementaryProblems, 109
2 Logic 115
2.1 Introduction, 116
2.2 PropositionalCalculus, 121
2.3 ArgumentsI, 146
2.4 ArgumentsII, 167
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viii CONTENTS
2.5 AShortRevisittoSetTheory, 171
2.6 BooleanAlgebra, 173
2.7 SupplementaryProblems, 177
3 Proofs 183
3.1 Introduction, 183
3.2 DirectProof, 193
3.3 IndirectProof, 212
3.4 MathematicalInduction, 218
3.5 SupplementaryProblems, 241
4 Functions 247
4.1 Introduction, 247
4.2 Relations, 248
4.3 Functions, 274
4.4 SupplementaryProblems, 321
5 GroupTheory 327
5.1 Introduction, 327
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5.3 Subgroups, 356
5.4 CyclicGroups, 382
5.5 HomomorphismsandIsomorphisms, 385
5.6 NormalSubgroups, 404
5.7 Centralizer,Normalizer,Stabilizer, 412
5.8 QuotientGroup, 419
5.9 TheIsomorphismTheorems, 427
5.10 DirectProductofGroups, 437
5.11 SupplementaryProblems, 441
6 LinearAlgebra 447
6.1 Introduction, 447
6.2 VectorSpace, 449
6.3 LinearDependenceandIndependence, 456
6.4 BasisandDimensionofaVectorSpace, 461
6.5 Subspaces, 469
6.6 LinearTransformations – LinearOperators, 477
6.7 IsomorphismofLinearSpaces, 489
6.8 LinearTransformationsandMatrices, 501
6.9 LinearSpaceM , 507
mn
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