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

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Then:

$\ds \map \complement {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \map \complement H$


Proof

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

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources