Question

For two finite disjoint sets $A$ and $B$, if the number of elements in power set of $A$ is $224$ more than the number of elements in power set of $B$, then the number of elements present in either of the sets is

• A. $12$
• B. $0$
• C. $15$
• D. $13$

Answer 1

The correct option is D $13$

Let $n\left(A\right)=x$ and $n\left(B\right)=y(x>y)$
Number of elements in $P\left(A\right)$ i.e., $n\left(P\right(A\left)\right)=2^x$
and $n\left(P\right(B\left)\right)=2^y$
Given that $2^x-2^y=224$
$\Rightarrow 2^y(2^{x-y}-1)=32\times 7\Rightarrow 2^y(2^{x-y}-1)=2^5(2^3-1)$
On comparison, we get
$y=5,x-y=3\Rightarrow x=8$

Since $A$ and $B$ are disjoint,
$\therefore A\cap B=\varphi \Rightarrow n(A\cap B)=0$
Hence, $n(A\cup B)=n\left(A\right)+n\left(B\right)=13$