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

## Theorem

Let $S$ be a set.

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

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T \subseteq \powerset T$.

Then:

#### Complement of Intersection

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

#### Complement of Union

$\ds \relcomp S {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \relcomp S H$

## Source of Name

This entry was named for Augustus De Morgan.