Question

Let A, B, and C be sets in a universal set U. We are given n(U) = 63, n(A) = 33, n(B) = 34, n(C) = 28, n(A ∩ B) = 15, n(A ∩ C) = 17, n(B ∩ C) = 14, n(A ∩ B ∩ CC) = 9. Find the following values.

(a) n(AC ∩ B ∩ C)

(b) n(A ∩ BC ∩ CC)

Answer 1

given,

n(U)=63, n(A)=33, n(B)=34, n(C)=28, n(AnB)=15, n(AnC)=17, n(BnC)=14, N(AnBnCC)=n(AnBnC)=9

we can calculate the followings first,

n(AuB) = n(A)+n(B)-n(AnB) = 33+34-15=52

n(AuC) = n(A)+n(C)-n(AnC) = 33+28-17=44

n(BuC) = n(B)+n(C)-n(BnC) = 34+28-14=47

n(AuBuC) = n(U) = 63

a.

(ACnBnC)=[ACn(BnC)]

,=AC+(BnC)-[ACu(BnC)] [by AnB=A+B-AuB]

=AC+BC-[AC + BC- ACnBC] [by AuB=A+B-AB and BnC=BC]

=AC+BC-AC-BC+ABC

=ABC=AnBnC

hence, n(ACnBnC)=n(AnBnC) = 9.

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b.

[AnBCnCC]=[(AnBC)nCC]

=AnBC+CC-[(AnBC)uCC]   [by AnB=A+B-AuB]

=ABC+CC-[(AuCC)n(BCuCC)] [by (AnB)uC=(AnC)u(BnC)]

=ABC+C-[(AuC)n(BCuC)] [CC=C]

=ABC+C-[(AuC)nC] [(BCuC)=C]

=ABC+C-C [(AuC)nC=C]

=ABC = AnBnC

n[AnBCnCC]=n(AnBnC)=9