De Morgan's Laws (Set Theory)/Relative Complement/General Case

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Theorem

Let $S$ be a set.

Let $T$ be a subset of $S$.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.


Then:

Complement of Intersection

$\displaystyle \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$


Complement of Union

$\displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$


Source of Name

This entry was named for Augustus De Morgan.