The power set

The power set ${\mathcal P}(S)$ of a set $S$ is the set of all its subsets. For instance, if $S=\{1,2,3\}$, then

$${\mathcal P}(S) = \{\emptyset,\{1\},\{2\},\{3\}, \{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\},$$

so $\char93 {\mathcal P}(S)=8$.

For any set $S$, there is a bijection between ${\mathcal P}(S)$ and the set $\{0,1\}^S$ of functions $\chi:S \rightarrow \{0,1\}$ that maps the subset $T$ of $S$ to its characteristic function

$$\chi_T(s):= \begin{cases} 1, \text{ if $s \in T$} \\ 0, \text{ if $s \not\in T$} \end{cases}.$$

Therefore $\char93 {\mathcal P}(S) = \char93 \left( \{0,1\}^S \right) = 2^{\char93 S}$.