Power set

A power set of a set $A$ is the set

Thus $\mathcal{P}(A)$ collects all subsets of $A$ into a single set . Power sets are central when forming collections of sets, such as families indexed by an index set.

Examples:

• If $A=\varnothing$, then $\mathcal{P}(A)=\{\varnothing\}$.
• If $A=\{1,2\}$, then $\mathcal{P}(A)=\{\varnothing,\{1\},\{2\},\{1,2\}\}$.