Question

If $A$ is the void set $\varphi$, then $P\left(A\right)$ has just one element $\varphi$ i.e. $P[\varphi ]=\left\{\varphi \right\}$. So number of elements of $P\left[P\left(P\left(\varphi \right)\right)\right]$ is

• A. $4$
• B. $3$
• C. $5$
• D. $2^4$

Answer 1

Answer: (A)

We have, $P(\varphi )=\left\{\varphi \right\}$
$\therefore P\left(P\left(\varphi \right)\right)=\left\{\varphi ,\left\{\varphi \right\}\right\}$
$\Rightarrow P\left[P\left(P\left(\varphi \right)\right)\right]=\left\{\varphi ,\left\{\varphi \right\},\left\{\left\{\varphi \right\}\right\},\left\{\varphi ,\left\{\varphi \right\}\right\}\right\}$
Hence, $P\left[P\left(P\left(\varphi \right)\right)\right]$ consists of $4$ elements i.e. $n\left[P\left[P\left(P\left(\varphi \right)\right)\right]\right]=4$