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

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## Theorem

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of sets, all of which are subsets of a universe $\mathbb U$.

Then:

#### Complement of Intersection

- $\displaystyle \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$

#### Complement of Union

- $\displaystyle \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$

## Source of Name

This entry was named for Augustus De Morgan.