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

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Theorem

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


Then:

$\displaystyle \complement \paren {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \complement \paren H$


Proof

\(\displaystyle \complement \paren {\bigcap \mathbb T}\) \(=\) \(\displaystyle \mathbb U \setminus \paren {\bigcap \mathbb T}\) $\quad$ Definition of Set Complement $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{H \mathop \in \mathbb T} \paren {\mathbb U \setminus H}\) $\quad$ De Morgan's Laws: Difference with Intersection $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{H \mathop \in \mathbb T} \complement \paren H\) $\quad$ Definition of Set Complement $\quad$

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources