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 the ordered pair $\left(n\right(A),n(B\left)\right)=$

• A. $(6,2)$
• B. $(8,4)$
• C. $(4,0)$
• D. $(0,1)$

Answer 1

The correct option is C $(4,0)$

Let ,
Number of element in subset A be $n\left(A\right)=m$ and
Number of element in subset B be $n\left(B\right)=n$
Then the total number of subsets of a finite 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 the equation we get
$2^m=2^n+15$
$2^m-2^n=15\Rightarrow m=4;n=0$