Relative Complement Mapping on Powerset is Bijection/Proof 1

Theorem

Let $S$ be a set.

Let $\complement_S: \powerset S \to \powerset S$ denote the relative complement mapping on the power set of $S$.

Then $\complement_S$ is a bijection.

Thus each $T \subseteq S$ is in one-to-one correspondence with its relative complement.

Proof

From Relative Complement of Relative Complement:

$\forall X \subseteq S: \relcomp S {\relcomp S X} = X$

That is, $\complement_S$ is an involution.

The result follows from Mapping is Involution iff Bijective and Symmetric.

$\blacksquare$