Question:
If A and B are two finite sets such that n(A) > n(B) and the difference of the number of elements of the power sets of A and B is 96, then n(A) – n(B) = ____________.
Solution:
If n(A) > n(B) and n(P(A)) – n(P(B)) = 96 given
where $P(A)$ and $P(B)$ represents power left of $A \neq B$ respectively.
Let n(A) = n and n(B) = m
i.e n(P(A)) = 2n and n(P(B)) = 2m
i.e 2n – 2m = 96
2m(2n –m – 1) = 96 = 25 × 3
i.e 2m = 25
i.e m = 5 and 2n –m – 1 = 3
2n –m = 4 = 22
i.e. n – m = 2
i.e n = 2 + m
n = 2 + 5
i.e. n = 7
∴ n(A) – n(B) = n – m = 2