Complement of Complement

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Theorem

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

$\complement \left({\complement \left({S}\right)}\right) = S$


Proof

Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $\complement_S \left({\complement_S \left({T}\right)}\right) = T$ from Relative Complement of Relative Complement.

$\blacksquare$


Sources