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EXAMPLES
Example1
Thefollowingsetsaregiven:A={1,2,3},B={1,2,4},C={1,2}andD=∅.
Findtheircardinalities.
Solution
ThesetAhasthreemembers:1,2i3.Itscardinalityisthen|A|=3.Similarly
|B|=3and|C|=2.Distheemptyset,soithasnomembers.Thus|D|=0.
Example2
Readthefollowingnotationandthenlistalltheelementsofthegivenset:
A
=
{
x
∈
N
:<
x
5
}
.
Solution
“Aisthesetofthenaturalnumbersxsuchthatxislowerthan5”.Itsmembers
are:0,1,2,3and4.
Example3
Readthefollowingnotationandthenlistalltheelementsofthegivenset:
X
=
{
x
:
−
10
<
x
<
10
∧
(
∃
y
∈
Z
:
x
=
3
y
)
}
.
Solution
“Xisthesetofxsuchthatxgreaterthan–10,xislowerthan10andthereexists
anintegerysuchthatxisequalto3y”.Twofirstconditionsaretrivialandthe
thirdonemeansthatxisdivisibleby3.Therearesevennumbersbeingmembers
ofX:–9,–6,–3,0,3,6and9.
Example4
Thefollowingsetsaregiven:A={1,2,3,4},B=(1,4),C=[1,4],D=N∩C,
E={x∈Z:x∈B}.Listalltherelationsofinclusionandequality.
Solution
ThesetAhasfourmembers:1,2,3and4.ThesetsBandCareintervalsfrom1
to4,withthedifferencethattheendsbelongtoC(closedinterval),butnottoB
(openinterval).ThesetDisintersectionofCandthesetofnaturalnumbersN,
soithasmembers1,2,3and4.SimilarlythesetEconsistsofthenumbers2and
3(onlythesetwonumbersaremembersofB–1and4aretheendsofthe
intervalandBisopen).
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