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

## Theorem

Let $S$ be a set.

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

Then:

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

## Proof

 $\displaystyle \relcomp S {\bigcap_{i \mathop \in I} S_i}$ $=$ $\displaystyle S \setminus \paren {\bigcap_{i \mathop \in I} S_i}$ Definition of Relative Complement $\displaystyle$ $=$ $\displaystyle \bigcup_{i \mathop \in I} \paren {S \setminus S_i}$ De Morgan's Laws for Set Difference: Difference with Intersection $\displaystyle$ $=$ $\displaystyle \bigcup_{i \mathop \in I} \relcomp S {S_i}$ Definition of Relative Complement

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.