Question

If $A=\varnothing ,P\left(A\right)$ denotes power set of $A$, then number of elements in $p\left(p\right(p\left(p\right(pA\left)\right)\left)\right))$ is

• A. 1
• B. $2^4$
• C. $2^5$
• D. $2^{16}$

Answer 1

Answer: (B)

($2^4$)

As we know that the number of elements in a power set $P\left(X\right)$ is $2^n$
Where $n$ is the number of elements in set $X$.
Given number of elements in $A$ is $0$ as $A$ is empty set.
So, number of elements in power set $P\left(A\right)$ is $n\left(A\right)=2^0=1$ [since, $n=0$]
Similar way, $n\left[P\right\{P\left(A\right)\left\}\right]=2^1=2$
$\Rightarrow n\left(P\right(P\left\{P\right(A\left)\right\}\left)\right)=2^2=4$
$\therefore P\left\{P\right\{P\left\{P\right(A\left)\right\}\left\}\right\}=2^4$
Hence, option $B$ is correct