Question

If A is a finite set, let P(A) denote the set of all subsets of A and n(A) 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(X) and n(Y).

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

Answer 1

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

A set containing n elements has $2^n$ subsets.

No. of subsets of X = No. of subsets of Y + 15.

No. of subsets of Y must be an odd no. if no. of elements in X is not 0 which obviously it is not. The only case when $2^n$ is odd is when $n=0$. hence n(Y)=0.

$n\left(P\right(X\left)\right)=16\Rightarrow 2^n=16\Rightarrow n=4$