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

## Theorem

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universe $\mathbb U$.

Then:

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

## Proof

 $\displaystyle \map \complement {\bigcup_{i \mathop \in I} S_i}$ $=$ $\displaystyle \mathbb U \setminus \paren {\bigcup_{i \mathop \in I} S_i}$ Definition of Set Complement $\displaystyle$ $=$ $\displaystyle \bigcap_{i \mathop \in I} \paren {\mathbb U \setminus S_i}$ De Morgan's Laws for Set Difference: Difference with Union $\displaystyle$ $=$ $\displaystyle \bigcap_{i \mathop \in I} \map \complement {S_i}$ Definition of Set Complement

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.