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

## 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:

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

## Proof

 $\displaystyle \complement_S \left({\bigcup \mathbb T}\right)$ $=$ $\displaystyle S \setminus \left({\bigcup \mathbb T}\right)$ Definition of Relative Complement $\displaystyle$ $=$ $\displaystyle \bigcap_{H \mathop \in \mathbb T} \left({S \setminus H}\right)$ De Morgan's Laws: Difference with Union $\displaystyle$ $=$ $\displaystyle \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$ Definition of Relative Complement

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.