1. 1.

   If $X$ is finite, then $|\mathcal{P}(X)|=2^{{|X|}}$.

2. 2.

   The above property also holds when $X$ is not finite. For a set $X$, let $|X|$ be the cardinality of $X$. Then $|\mathcal{P}(X)|=2^{{|X|}}=|2^{X}|$, where $2^{X}$ is the set of all functions from $X$ to $\{0,1\}$.

3. 3.

   For an arbitrary set $X$, Cantor’s theorem states: a) there is no bijection between $X$ and $\mathcal{P}(X)$, and b) the cardinality of $\mathcal{P}(X)$ is greater than the cardinality of $X$.

Suppose $S=\{a,b\}$. Then $\mathcal{P}(S)=\{\emptyset,\{a\},\{b\},S\}$. In particular, $|\mathcal{P}(S)|=2^{{|S|}}=4$.

If $X$ is a set, then the finite power set of $X$, denoted by $\mathcal{F}(X)$, is the set whose elements are the finite subsets of $X$.

Due to the canonical correspondence between elements of $\mathcal{P}(X)$ and elements of $2^{X}$, the power set is sometimes also denoted by $2^{X}$.