Finite Mathematics
m
TSILB1
Version 4.0A0, 5 October 1998
m m m m
1
ThismSpacemIntentionallymLeftmBlank.m Contributorsminclude:mJohnmG.m
Ke-
mmeny,mJ.mLauriemSnell,mandmGeraldmL.mThompson. mAdditionalm workm by:mP
etermDoyle.m Copyrightm(C)m1998mPeterm G.mDoyle.m Derivedm frommworksmC
opyright
(C)m1957,m1966,m1974mJohnmG.mKemeny,mJ.mLauriemSnell,mGeraldmL.mThom
p-
mson.m Thism workmism freelymredistributablem underm themtermsmofmthem GNUm
FreemDocumentationmLicense.
,2
,Chapter 2 mm
Sets and subsets
m m
2.1 Introduction
Amwell-
definedmcollectionmofmobjectsmismknownmasmamset.mThismconcept,minmitsmc
ompletemgenerality,mismofmgreatmimportanceminmmathematicsmsincemallm
ofmmathematicsmcanmbemdevelopedmbymstartingmfrommit.
Them variousm piecesm ofm furniturem inm am givenm roomm formm am set.m So
m domthembooksminmamgivenmlibrary,mormthemintegersmbetweenm1mandm1,000,
000,mormallmthemideasmthatmmankindmhasmhad,mormthemhumanmbeingsmalivem
betweenmonem billionm B.C.m andm tenm billionm A.D.m Thesem examplesm arem al
lm examplesmofm finitem sets,m thatm is,m setsm havingm am finitem numberm ofm ele
ments.m Allm themsetsm discussedm inm thism bookm willm bem finitem sets.
Therem arem twom essentiallym differentm waysm ofm specifyingm am set.m On
emcanmgivemamrulembymwhichmitmcanmbemdeterminedmwhethermormnotmamgive
nmobjectmismammembermofmthemset,mormonemcanmgivemamcompletemlistmofmthe
melementsm inm them set.m Wem shallm saym thatm them formerm ism am description
m ofmthem setm andm them latterm ism am listingm ofm them set.m Form example,m wem ca
nm definemamsetmofmfourmpeoplemasm(a)mthemmembersmofmthemstringmquartetm
whichm playedm inm townm lastm night,m orm (b)m fourm particularm personsm who
sem namesmarem Jones,m Smith,m Brown,m andm Green.m Itm ism customarym tom
usem bracesm to,m surroundm them listingm ofm am set;m thusm them setm abovem s
houldm bem listed
{Jones,m Smith,m Brown,m Green}.
Wemshallmfrequentlymbeminterestedminmsetsmofmlogicalmpossibilities,msin
cemthemanalysismofmsuchmsetsmismverymoftenmammajormtaskminmthemsolving
mofmamproblem.m Suppose,mformexample,mthatmwemwereminterestedminmthems
uccessesmofmthreemcandidatesmwhomentermthempresidentialmprimariesm(
wemassumemtheremaremnomothermentries).m Supposemthatmthemkeymprimar
iesmwillmbemheldminm Newm Hampshire,m Minnesota,m Wisconsin,m andm Ca
lifornia.m m Assume
, 3
m
TSILB1
Version 4.0A0, 5 October 1998
m m m m
1
ThismSpacemIntentionallymLeftmBlank.m Contributorsminclude:mJohnmG.m
Ke-
mmeny,mJ.mLauriemSnell,mandmGeraldmL.mThompson. mAdditionalm workm by:mP
etermDoyle.m Copyrightm(C)m1998mPeterm G.mDoyle.m Derivedm frommworksmC
opyright
(C)m1957,m1966,m1974mJohnmG.mKemeny,mJ.mLauriemSnell,mGeraldmL.mThom
p-
mson.m Thism workmism freelymredistributablem underm themtermsmofmthem GNUm
FreemDocumentationmLicense.
,2
,Chapter 2 mm
Sets and subsets
m m
2.1 Introduction
Amwell-
definedmcollectionmofmobjectsmismknownmasmamset.mThismconcept,minmitsmc
ompletemgenerality,mismofmgreatmimportanceminmmathematicsmsincemallm
ofmmathematicsmcanmbemdevelopedmbymstartingmfrommit.
Them variousm piecesm ofm furniturem inm am givenm roomm formm am set.m So
m domthembooksminmamgivenmlibrary,mormthemintegersmbetweenm1mandm1,000,
000,mormallmthemideasmthatmmankindmhasmhad,mormthemhumanmbeingsmalivem
betweenmonem billionm B.C.m andm tenm billionm A.D.m Thesem examplesm arem al
lm examplesmofm finitem sets,m thatm is,m setsm havingm am finitem numberm ofm ele
ments.m Allm themsetsm discussedm inm thism bookm willm bem finitem sets.
Therem arem twom essentiallym differentm waysm ofm specifyingm am set.m On
emcanmgivemamrulembymwhichmitmcanmbemdeterminedmwhethermormnotmamgive
nmobjectmismammembermofmthemset,mormonemcanmgivemamcompletemlistmofmthe
melementsm inm them set.m Wem shallm saym thatm them formerm ism am description
m ofmthem setm andm them latterm ism am listingm ofm them set.m Form example,m wem ca
nm definemamsetmofmfourmpeoplemasm(a)mthemmembersmofmthemstringmquartetm
whichm playedm inm townm lastm night,m orm (b)m fourm particularm personsm who
sem namesmarem Jones,m Smith,m Brown,m andm Green.m Itm ism customarym tom
usem bracesm to,m surroundm them listingm ofm am set;m thusm them setm abovem s
houldm bem listed
{Jones,m Smith,m Brown,m Green}.
Wemshallmfrequentlymbeminterestedminmsetsmofmlogicalmpossibilities,msin
cemthemanalysismofmsuchmsetsmismverymoftenmammajormtaskminmthemsolving
mofmamproblem.m Suppose,mformexample,mthatmwemwereminterestedminmthems
uccessesmofmthreemcandidatesmwhomentermthempresidentialmprimariesm(
wemassumemtheremaremnomothermentries).m Supposemthatmthemkeymprimar
iesmwillmbemheldminm Newm Hampshire,m Minnesota,m Wisconsin,m andm Ca
lifornia.m m Assume
, 3
