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

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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:

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


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.


Sources