= Power set
{title2=$\mathcal P(X)$}
{wiki}
The power set $\mathcal P(X)$ is the <set> of all <subsets> of $X$, including the <empty set> and $X$ itself. It is useful whenever an object is a selection rather than an individual member: a possible collection of active vertices in a <graph>, for example, belongs to the <power set> of the vertex <set>.
For $X=\{a,b,c\}$, the <power set> has eight members:
$$
\mathcal P(X)=\{\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}.
$$
Each of the three <set elements> is either included or excluded. More generally, if $|X|=n<\infty$, these $n$ independent binary choices give $|\mathcal P(X)|=2^n$. This is a <bijection> between <subsets> and <binary strings> of length $n$, not a probabilistic independence assumption.
\Image[/power-set-inclusion-cube.png]
{title=Eight subsets of a three-element set form a cube when edges add one element}
Ordering the <subsets> by inclusion gives a <Boolean lattice>. In the picture an upward edge adds one member. Complementation reverses inclusion, and <union> and <intersection> combine selections. The empty-input case is worth checking: $\mathcal P(\varnothing)=\{\varnothing\}$ has one member, not zero, consistently with $2^0=1$.
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