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

Theorem

Let $S$ be a set.

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

Then:

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

Proof

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

$\blacksquare$

Source of Name

This entry was named for Augustus De Morgan.