Question

If X is a finite set. Let P(X) denote the set of all subsets of X and let n(X) denote the number of elements in X. If for two finite subsets A, B, n(P(A)) = n(P(B)) + 15 then n(B) = and n(A) =

• A. $n\left(A\right)=4,n\left(B\right)=0$
• B. $n\left(A\right)=0,n\left(B\right)=0$
• C. $n\left(A\right)=4,n\left(B\right)=4$
• D. $n\left(A\right)=0,n\left(B\right)=4$

Answer 1

The correct option is A $n\left(A\right)=4,n\left(B\right)=0$

Let number of element in subset A be n(A) = m

Number of element in subset B be n(B) = n

Than total number of subset of finite set containing say n elements in 2${}^n$

$\therefore n\left[P\right(A\left)\right]=2^m,n\left[P\right(B\left)\right]=2^n$

Substituting in the equation we get

$2^m=2^n+152^m-2^n=15\Rightarrow m=4;n=0$

$m=n\left(A\right)=4;n=n\left(B\right)=0$

Option A is correct answer.