Relative Complement of Relative Complement

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Theorem

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

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

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \complement_S \left({\complement_S \left({T}\right)}\right)$	$=$	$\displaystyle S \setminus \left({S \setminus T}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of relative complement
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle S \cap T$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Set Difference with Set Difference

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.

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \complement_S \left({\complement_S \left({T}\right)}\right)\)	\(=\)	\(\displaystyle S \setminus \left({S \setminus T}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of relative complement
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle S \cap T\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Set Difference with Set Difference