Power Set is Closed under Set Complement

Theorem
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
 * $\forall A \in \powerset S: \relcomp S A \in \powerset S$

Proof
Let $A \in \powerset S$.

Then by the definition of power set, $A \subseteq S$.

By definition of relative complement:
 * $\relcomp S A = \set {x \in S: x \notin A}$

Hence $\relcomp S A$ is a subset of $S$.

That is:
 * $\relcomp S A \in \powerset S$

and closure is proved.

Also see

 * Power Set is Closed under Union
 * Power Set is Closed under Intersection
 * Power Set is Closed under Set Difference