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denoteitbyA⊄B.Moreformally,inclusionofAinBcanbedefinedas
follows:
A
⊂
B
⇔
§
¨
©
∀
x
x
∈
A
3
x
∈
B
·
¸
¹
.
Note
:inclusionofonesetinanotherdoesnotmeanthattheyareequal!Inorder
toobtainequality,mutualinclusionmustoccur.Asanexample,letusconsider
thefollowingsets:A={1,2,3}andB={1,2}.InthiscaseB⊂A,becauseall
themembersofB(thatis“1”and“2”)belongalsotoA.Ontheotherhand“3”
belongstoAanddoesnotbelongtoB,soA⊄B.
Rememberthatif
A=B,thenbothinclusionrelationsmustoccur:A⊂B
andB⊂A.Inparticular,foranysetAitistruethatA⊂A.
Aspecialtypeofsetisthe
emptyset
,denotedbythesymbol∅.Emptyset
has
nomembers
.ItfollowsinparticularthatforanysetAthecondition∅⊂Ais
satisfied.Sincetheemptysetcontainsnoelements,thesentence“x∈∅”isfalse
foreveryelementx.Forthisreason,youshouldavoidsuchnotation,for
example,whensolvingequationsorinequalities.
Kindoftheoppositeoftheemptysetisthe
universe
(otherwiseknownasthe
space),beingthesetthatcontainsalltheobjectsthatweareinterestedinatthe
moment.TheuniverseisdenotedbytheletterU.Usuallytheformofthe
universeresultsfromthecontext.ForeverysetAitistruethatA⊂U.
Complement
ofasetA,denotedbyA′(orA
C
),isthesetofalltheelementsnot
belongingtoA.Inotherwords,thecomplementofAmaybedescribedlikethis:
A
′
=
{
x
:
x
∉
A
}
orlikethis:
x
∈
A
′
⇔
¬
x
∈
A
.Graphicalinterpretationofthe
complementisshowninFig.2.1.
A
A′
U
Fig02010Complementofaset
23
