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) = ____________.

Answer 1

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 ≠ B respectively.
Let n(A) = n and n(B) = m
i.e n(P(A)) = 2ⁿ and n(P(B)) = 2^m
i.e 2ⁿ – 2^m = 96
2^m(2^{n –m} – 1) = 96 = 2⁵ × 3
i.e 2^m = 2⁵
i.e m = 5 and 2^{n –m} – 1 = 3
2^{n –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