Relative Complement of Relative Complement

Theorem
The relative complement of the relative complement of a set is itself:


 * $\complement_S \left({\complement_S \left({T}\right)}\right) = T$

Thus, considered as a mapping on the power set of $S$:
 * $\complement_S: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right): \complement_S \left({T}\right) = S \setminus T$

$\complement_S$ is an involution.

Proof
The definition of the relative complement requires that $T \subseteq S$.

But we have $T \subseteq S \iff T \cap S = T$ from Intersection with Subset is Subset‎.

Thus $\complement_S \left({\complement_S \left({T}\right)}\right) = T$ follows directly.