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
Let
\[ n(A)=m,\qquad n(B)=n \]
Number of elements in power sets:
\[ n(P(A))=2^m,\qquad n(P(B))=2^n \]
Given,
\[ 2^m-2^n=96 \]
\[ 2^n(2^{m-n}-1)=96 \]
Since
\[ 96=2^5 \times 3 \]
Taking
\[ 2^n=2^5 \]
we get
\[ 2^{m-n}-1=3 \]
\[ 2^{m-n}=4 \]
\[ m-n=2 \]
Therefore,
\[ n(A)-n(B)=2 \]
Answer
\[ \boxed{2} \]