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

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## Theorem

Let $S$ be a set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Then:

#### Complement of Intersection

- $\ds \relcomp S {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \relcomp S {S_i}$

#### Complement of Union

- $\ds \relcomp S {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$

## Source of Name

This entry was named for Augustus De Morgan.

## Sources

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- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (next): $2.22$