Question

If $P\left(A\right)$denotes the power set of $A$and $A$ is the void set, then what is the number of elements in $P\left\{P\right\{P\{P\left(A\right)\left\}\right\}\}$?

• A. $0$
• B. $1$
• C. $4$
• D. $16$

Answer 1

The correct option is D

$16$

Explanation for the correct option.

Step 1: Apply this formula:

If set $A$ has $n$ elements then $P\left(A\right)$ has $2^n$ elements.

Since $A$ is void set it means that there is no element in set $A$.

Hence, the number of elements in the power set of $A=2^0=1.$

Step 2: Put the different values of $P\left(A\right)$

When $P\left(A\right)$ has $1$ element.

$$\begin{gathered} \Rightarrow P\{P\left(A\right)\}=2^1 \\ =2\mathrm{elements} \end{gathered}$$

When $P\left\{P\right(A\left)\right\}$ has $2$ elements.

$$\begin{gathered} \Rightarrow P\left\{P\right\{P\left(A\right)\left\}\right\}=2^2 \\ =4\mathrm{elements} \end{gathered}$$

When $P\left\{P\right\{P\left(A\right)\left\}\right\}$ has $4$ elements.

$$\begin{gathered} P\left\{P\right\{P\{P\left(A\right)\left\}\right\}\}=2^4 \\ =16\mathrm{elements} \end{gathered}$$

Hence, option D is correct.