Question

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

• A. $n\left(A\right)=4,n\left(B\right)=0$
• B. $n\left(A\right)=5,n\left(B\right)=0$
• C. $n\left(A\right)=4,n\left(B\right)=2$
• D. $n\left(A\right)=6,n\left(B\right)=0$

Answer 1

Answer: (A)

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

If X is a finite set.

Let,

Number of element in a subset $A$ be $n\left(A\right)$ =$m$ and

Number of elements in subset $B$ be $n\left(B\right)$=$n$

Then total number of subsets of finite set contaning say $n$ elements is $2^n$.

Therefore,

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

Substituting in equations , we get:-

$2^m=2^n+15$

$2^m-2^n=15$

$m=4,n=0$

Option $\text{A}$ is correct