De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets

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Theorem

Let $S$ be a set.

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.


Then:

Complement of Intersection

$\displaystyle \relcomp S {\bigcap_{i \mathop \in I} \mathbb S_i} = \bigcup_{i \mathop \in I} \relcomp S {S_i}$


Complement of Union

$\displaystyle \relcomp S {\bigcup_{i \mathop \in I} \mathbb S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$


Source of Name

This entry was named for Augustus De Morgan.


Sources