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

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Theorem

Let $\mathbb T$ be a set of sets, all of which are subsets of a universe $\mathbb U$.


Then:

$\displaystyle \complement \paren {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \complement \paren H$


Proof

\(\displaystyle \complement \paren {\bigcup \mathbb T}\) \(=\) \(\displaystyle \mathbb U \setminus \paren {\bigcup \mathbb T}\) Definition of Set Complement
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{H \mathop \in \mathbb T} \paren {\mathbb U \setminus H}\) De Morgan's Laws for Set Difference: Difference with Union
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{H \mathop \in \mathbb T} \complement \paren H\) Definition of Set Complement

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources