Power Sets

Given a set $S$, the power set of $S$ is the set of all subsets of the set $S$. The power set of $S$ is denoted as $\mathcal{P}(S)$

For example

$$\begin{array}{c}\mathcal{P}\left(\left\{0,1,2\right\}\right)=\left\{\varnothing ,\left\{0\right\},\left\{1\right\},\left\{2\right\},\left\{0,1\right\},\left\{0,2\right\},\left\{1,2\right\},\left\{0,1,2\right\}\right\}\\ \mathcal{P}(\varnothing )=\left\{\varnothing \right\}\\ \mathcal{P}(\left\{\varnothing \right\})=\left\{\varnothing ,\left\{\varnothing \right\}\right\}\end{array}$$

If $\left|S\right|=n$, than we have $\left|\mathcal{P}(S)\right|=2^n$