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CHAPTER2
SETS
2010BASICCONCEPTS
THEORYINANUTSHELL
Set
ismostoftenclassifiedasoneofthe
primitivenotions
,thatisundefined.
Forourpurposes,wecancharacterizesetasacollectionof
uniqueandnot
orderedelements
3
.
Wedenotethesetsbycapitalletters:A,B,C,ł,andtheircomponents(called
members
)–bythelowercaseletters.Thefactthattheelementxbelongstothe
setA,iswrittenas:x∈A,whilethefactthatitdoesn’t:x∉A.
ThenumberofmembersofAisdenotedby|A|andiscalledthecardinalityofA.
Ifitisfinite,sometimesitisdenotedbyn(A).
Setsmaybedescribedinvariousways.Inthisbook,wewillbemostly
interestedinfinitesetsandsetsofnumbers.Therefore,torepresentthemembers
ofasetwewilluse:
•
enumerationofallthemembersoftheset,forexampleA={1,2,3,4};
•
notationusingformulae,inequalities,equations,etc.,forexample:
A={x:x∈N∧x
ł
1∧x
ś
4}(“:”–colon–means“suchthat”;given
expressionmeanthus“Aisthesetofxsuchthatxisanaturalnumber,x
isnotlowerthan1andxisnotgreaterthan4”;notethatthesetdefined
thiswayisexactlythesameasthesetfromthepreviousbulletpoint);
•
notationusingintervals(seelaterinthischapter),forexample:A=[1,4]
(observethatthistimethesetisnotsameasbefore,asforinstance3.5is
itsmember).
The
equality
oftwosetsAandBmeansthattheyhaveexactlythesame
members.WewriteitasA=B.Moreformally,equalityoftwosetscanbe
definedasfollows:
A
=
B
⇔
§
¨
©
∀
x
x
∈
A
⇔
x
∈
B
·
¸
¹
.
The
inclusion
oftwosetsAandBmeansthatallthemembersofonesetarealso
themembersofanotherone.InclusionofAinBisdenotedbyA⊂BorA⊆B.
InsuchacasewesaythatA
issubsetof
B.IfAisnotsubsetofB,thenwe
3
Inthecaseofrepetitionsofelementswearetalkingaboutmultisets.Inthecaseof
orderedelementswearetalkingaboutsequences.Thislastconceptisdiscussedindetail
laterinthebook.
22
