Editing Power set (section)

== Recursive definition ==
If {{math|''S''}} is a [[finite set]], then a [[recursive definition]] of {{math|{{itco|{{mathcal|P}}}}(''S'')}} proceeds as follows:
* If {{math|1=''S'' = {{mset}}}}, then {{math|1={{itco|{{mathcal|P}}}}(''S'') = {{mset| {{mset}} }}}}.
* Otherwise, let {{math|''e'' ∈ ''S''}} and {{math|1=''T'' = ''S'' &setminus; {{mset|''e''}}}}; then {{math|1={{itco|{{mathcal|P}}}}(''S'') = {{itco|{{mathcal|P}}}}(''T'') ∪ {{mset|''t'' ∪ {{mset|''e''}} : ''t'' ∈ {{itco|{{mathcal|P}}}}(''T'')}}}}.

In words: 
* The power set of the [[empty set]] is a [[singleton (mathematics)|singleton]] whose only element is the empty set.
* For a non-empty set {{math|''S''}}, let <math>e</math> be any element of the set and {{math|''T''}} its [[relative complement]]; then the power set of {{math|''S''}} is a [[union (set theory)|union]] of a power set of {{math|''T''}} and a power set of {{math|''T''}} whose each element is expanded with the {{math|''e''}} element.