Let A,B,C⊆U. If n(U)=54, n(A)=16, n(B)=24, n(C)=25, n(A∩B)=11, n(A∩C)=4, n(B∩C)=7, and n(A′∩B′∩C′)=8, find n(A′∩B′∩C).

Answer 1

Note that there are eight equations and eight unknowns (in or not in A, B, and C), so one way to attack this problem would be to set up a system of equations for the eight unknowns and solve.

However, if we want to construct a solution in a different way, note that

n(A'B'C) = n(A'B') - n(A'B'C')

n(A'B') = n(U) - n(A U B) = n(U) - (n(A) + n(B) - n(AB)) = n(U) + n(AB) - n(A) - n(B) = 54 + 11 - 16 - 24 = 25

n(A'B'C') = 25 - 8 = 17