An Introduction to Abstract Mathematics

345 Pages·2007·14.3 MB·English

by  Robert J. Bond

An Introducttoi on
AbstraMcatt hematics

An Introducnt tioo
AbstraMcatt hematics
J.
RobertB ond
BostoCno llege
J.
WilliamK eane
BostoCno llege
WAVElAND
PRESS,I NC.
Long GroveIl,lin ois
Fori nformataiboonu tth ibso okc,o ntact:
WavelanPdr esIsn,c .
4180I LR out8e3 ,Su it1e0 1
LongG roveI,L 60047-9580
(8476)3 4-0081
[email protected]
www .waveland.com
Copyrig©h 1t9 99b yR oberJtB. o nda ndW illiaJmK. e ane
Reissu2e0d0 7b yW avelanPdr esIsn,c .
lO-diIgSiBtN 1 -57766-539-2
13-diIgSiBtN 9 78-1-57766-539-7
Allr igrhetsse Nrvoep da.or ftt hbioso mka yb er eprodsutcoeirdnae,r d e trsyisetveaomlr,
transmiinat nftyoe rmdo rby anmye ans withouitnw rpietfrriomnmtig hs pesui bolni sher.
Printiendt heU niteSdt atoefsA merica
7 6 5 4 3
ToA nn andS arahf,o rt heilro vaen dc onstasnutp pofrotr
whata tt imesse emeadn e ndlejsosu rney.
To Liza ndM attf,o rt hefi rstti mea,n dt oF rancael,w ays.

CONTENTS
MathematiRceaals oning
1.1 Statement2s
1.2 CompoundS tatement1s6
1.3 Implicatio2n9s
1.4 ContraposiatnidvC eo nverse3 8
2 Sets 49
2.1 Setasn dS ubsets4 9
2.2 CombininSge ts 61
2.3 CollectioofSn est s 72
1 Functions 81
3.1 Definitiaonnd B asiPcr operti8e1s
3.2 SurjectainvdeI njectFiuvnec tion9s7
3.3 CompositiaonndI nvertiFbulnec tion1s1 0
4 BinarOyp eratioannsd R elations 123
4.1 BinarOyp eration1s2 3
4.2 EquivaleRneclea tion1s3 9
vii
viii CONTENTS
5 The Integers 151
5.1 Axiomsa ndB asiPcr opertie1s5 1
5.2 Inductio1n5 9
53. TheD ivisiAolng oritahnmd G reateCsotm mon
Divisor1s7 5
5.4 Primeasn dU niquFea ctorizat1i8o2n
5.5 Congruence1s8 9
5.6 Generalai Tzhienogr em 200
6 In fini teS ets 209
6.1 CountabSleet s 210
6.2 UncountabSleet sC,a ntorT'hse orema,n dt he
Schroeder-BernTshteeoirne m 220
6.3 CollectioofSn est s 229
7 The Reala ndC omplex Numbers 235
7.1 Fields2 35
7.2 TheR ealN umbers 243
7.3 TheC omplex Numbers 251
8 Poylnomiasl 263
8.1 Polynomial2s6 3
8.2 UniquFea ctorizat2i7o3n
8.3 PolynomioavlesrC ,R ,a ndQ 285
Answerasn dH inttsoS elecEtxeedr cis2e9s5
Bibliograp31h7y
Index3 19
PREFAFCOER
THEI NSTRUCTOR
Thibso oekv olfvreodam c ourtsheah ta bse etna ugahtBt o stCoonl lege
fomra nyye airnsm andyi ffefroer.nmT tsh ec ourtsaek,pe rni mabryi ly
sophommoarteh emmaatjiochrsas as,l wabyesei nn tentdope rde ptahroes e
studefnottrhs "e a bstmraatchte maotfti hctesi "t floetr,h r ei gcoarr,e ful
argumaenndlt o,g ipcraelc itshiaaotln,t homuegrheg llyi mpisnce adl culus,
woulbdel inchopfai lnflus r tshteu.rMd oysstt udewnihtlaslv hea tdw oor
thrseeem esotfce arlsc aunladus se mesotfle irn aelagre sbotr haat th ewyi ll
havaet taianr eeda sondaebglroeef me a themastoipchails tbiecfaotrieo n
startthibinosgo . k
Ourc hoiocfte o piimsco st ivbaytt ehfdea tchtas tt udteanktiesn troduc­
torcyo urisnae bss tarlagcetab nrdra e aaln aliynst ihsej iurn ioorsr e nior
yeaarssm athemamtaijco.Isr n so rdteorb ea blteoh andtlhee csoeu rses
effecttihveneyel eytd,ok notwh reu loefls o gaincdt hreu dimoefns test
theoarswy e lalss ombea spirco peorftf iuensc tbieotnwses eenti sn:j ective
ansdu rjefcutnicvteii moangasen,id n veirmsaega en,id n verftuinbclt.ei ons
Theb oodki vindaetsu rianlttlowy o( overlatpobp esi unrgpe,a) r. t s
• Chapt1e-r5is n trotdhufecu en dameonfta ablsst rmaactth ematics.
Logiscet,th eorreyl,a tfiuonncst,ai noodnp se,r atlieoatndoas c aresftuuld y
ofo naex isoyms ttehmie,n tegTehrescs.he a ptfeorrtsmh ceo roeft hbeo ok.
Wea lwacyosv tehrmi ast eriinoa ulcr o urasnewd,e u siett o i ntrooduurc e
studetnott hsle a nguoafmg aet hemaatnidtc ops r eptahreefm o trh eir
undergrcaadrueaietnmre as t hematics.
• Chapt6e-r8as p ptlhyie d eaansdt echnioqftu heeesa rlmiaetre .r ial
Thesceh aptfeotrrhs me o spta ratr ien depenadnedwn etc ,h oowshei ch
onetsoc ovaesrt impee rmaintdis n terleesaTtdhs.e t opitchso,u agrhe,
onetsh awtef eeolus rt udesnhtosuk lndo ww,i nlolfit n dc ompleatleileyn ,
ix