Find n(A’∪B) Finite

Question

Suppose that A and B are subsets of a universal set U. Suppose that:

N(U)=50

N(A)=10

n(B)=38

n(A’ ∩ B’)= 15

Find n(A’∪B)

Answer 1

Note that n(A')=n(U)- n(A)=50-10=40.

A'=A'∩U=A'∩(BUB')=(A'∩B) U (A'∩B'), where A'∩B and A'∩B' are disjoint since B and B' are disjoint.

Therefore, n(A')=n((A'∩B) U (A'∩B')) = n(A'∩B) + n(A'∩B') which gives 40 = n(A'∩B) + 15; so n(A'∩B)=25.

Now by the inclusion and exclusion principle, we have

n(A'UB)=n(A')+n(B)-n(A'∩B) = 40+38-25=53.