Proofs and Fundamentals

378 Pages·2011·4.667 MB·English

by  Bloch, Ethan D

Undergraduate Texts in Mathematics
Editorial Board
S. Axler
K.A. Ribet
For other titles Published in this series, go to
www.springer.com/series/666

Ethan D. Bloch
Proofs and Fundamentals
A First Course in Abstract Mathematics
Second Edition
Ethan D. Bloch
Mathematics Department
Bard College
Annandale-on-Hudson, NY 12504
USA
[email protected]
Editorial Board
S. Axler K.A.Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
ISSN 0172-6056
ISBN 978-1-4419-7126-5 e-ISBN 978-1-4419-7127-2
DOI 10.1007/978-1-4419-7127-2
Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011921408
© Springer Science+Business Media, LLC 2011
Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY
10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection
withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
Springer is part of Springer Science+Business Media (www.springer.com)
DedicatedtomywifeNancy,inappreciationofherloveandsupport

Contents
PrefacetotheSecondEdition........................................ xi
PrefacetotheFirstEdition .......................................... xiv
TotheStudent ..................................................... xix
TotheInstructor ...................................................xxiii
PartI PROOFS
1 InformalLogic................................................. 3
1.1 Introduction ............................................... 3
1.2 Statements ................................................ 4
1.3 RelationsBetweenStatements................................ 15
1.4 ValidArguments ........................................... 25
1.5 Quantifiers ................................................ 34
2 StrategiesforProofs............................................ 47
2.1 MathematicalProofs—WhatTheyAreandWhyWeNeedThem... 47
2.2 DirectProofs .............................................. 53
2.3 ProofsbyContrapositiveandContradiction..................... 57
2.4 Cases,andIfandOnlyIf .................................... 64
2.5 QuantifiersinTheorems ..................................... 70
2.6 WritingMathematics ....................................... 80
PartII FUNDAMENTALS
3 Sets........................................................... 91
3.1 Introduction ............................................... 91
3.2 Sets—BasicDefinitions ..................................... 93
viii Contents
3.3 SetOperations ............................................. 101
3.4 FamiliesofSets............................................ 109
3.5 AxiomsforSetTheory ...................................... 115
4 Functions ..................................................... 129
4.1 Functions ................................................. 129
4.2 ImageandInverseImage .................................... 140
4.3 CompositionandInverseFunctions ........................... 146
4.4 Injectivity,SurjectivityandBijectivity ......................... 154
4.5 SetsofFunctions........................................... 164
5 Relations...................................................... 171
5.1 Relations ................................................. 171
5.2 Congruence ............................................... 177
5.3 EquivalenceRelations....................................... 185
6 FiniteSetsandInfiniteSets...................................... 195
6.1 Introduction ............................................... 195
6.2 PropertiesoftheNaturalNumbers ............................ 196
6.3 MathematicalInduction ..................................... 201
6.4 Recursion ................................................. 212
6.5 CardinalityofSets.......................................... 221
6.6 FiniteSetsandCountableSets................................ 231
6.7 CardinalityoftheNumberSystems............................ 240
PartIII EXTRAS
7 SelectedTopics................................................. 251
7.1 BinaryOperations .......................................... 251
7.2 Groups ................................................... 257
7.3 HomomorphismsandIsomorphisms........................... 265
7.4 PartiallyOrderedSets....................................... 270
7.5 Lattices................................................... 280
7.6 Counting:ProductsandSums ................................ 288
7.7 Counting:PermutationsandCombinations ..................... 297
7.8 LimitsofSequences ........................................ 312
8 Explorations................................................... 323
8.1 Introduction ............................................... 323
8.2 GreatestCommonDivisors .................................. 324
8.3 DivisibilityTests ........................................... 326
8.4 Real-ValuedFunctions ...................................... 326
8.5 IterationsofFunctions ...................................... 327
8.6 FibonacciNumbersandLucasNumbers ....................... 328
8.7 FuzzySets ................................................ 330
Contents ix
8.8 YouAretheProfessor....................................... 332
Appendix:PropertiesofNumbers .................................... 341
References......................................................... 345
Index ............................................................. 351