Question

If A is a finite set, let $P\left(A\right)$ denote the set all subsets of A and $n\left(A\right)$ denote the number of elements in A. If for two finite sets X and Y, $n\left[P\right(X\left)\right]=n\left[P\right(Y\left)\right]+15$ then find $n\left(X\right)$ and $n\left(Y\right)$

• A. $n\left(X\right)=4;n\left(Y\right)=0$
• B. $n\left(X\right)=5;n\left(Y\right)=0$
• C. $n\left(X\right)=6;n\left(Y\right)=0$
• D. $n\left(X\right)=7;n\left(Y\right)=0$

Answer 1

The correct option is A $n\left(X\right)=4;n\left(Y\right)=0$

If $X$ is a finite set.

Let,

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

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

Then total number of subsets of finitr set containing 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