Question

The cardinality of the set $P\left\{P\right[P(\varnothing )\left]\right\}$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is D

$4$

Finding the cardinality of the set:

The cardinality of a set is defined as the number of elements in the set.

If set $A$ has $n$ elements then $P\left(A\right)$ has $2^n$ elements. So, the cardinality of $A$ is $2^n$.

We know the number of elements in the empty set is $0$.

$$\begin{gathered} \therefore P\left(\varnothing \right)=2^0 \\ =1 \end{gathered}$$

Now,

$$\begin{gathered} P\left[P\left(\varnothing \right)\right]=P\left(1\right) \\ =2^1 \\ =2 \end{gathered}$$

Therefore,

$$\begin{gathered} P\left\{P\right[P(\varnothing )\left]\right\}=P\left(2\right) \\ =2^2 \\ =4 \end{gathered}$$

Hence. the correct answer is option (D).