Question

For any empty set $X,$ the value of $n\left(P\right(P\left(P\right(P\left(P\right(X\left)\right)\left)\right)\left)\right)=$

• A. $2^4$
• B. $2^5$
• C. $2^6$

Answer 1

The correct option is A $2^4$

$X$ is an empty set.
$\Rightarrow P\left(X\right)=\left\{\varnothing \right\}$
$\Rightarrow n\left(P\right(X\left)\right)=2^0=1$
Now, $P\left(P\right(X\left)\right)=\{\varnothing ,\{\varnothing \left\}\right\}$
$\Rightarrow n\left(P\right(P\left(X\right)\left)\right)=2^1=2$
Similarly, repeating the same 5 times
$n\left(P\right(P\left(P\right(P\left(P\right(X\left)\right)\left)\right)\left)\right)=2^4=16.$