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

## Theorem

Let $S, T_1, T_2$ be sets such that $T_1, T_2$ are both subsets of $S$.

Then, using the notation of the relative complement:

$\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$

## Proof

Let $T_1, T_2 \subseteq S$.

Then from Union is Smallest Superset:

$T_1 \cup T_2 \subseteq S$

Hence:

 $\displaystyle \relcomp S {T_1 \cup T_2}$ $=$ $\displaystyle S \setminus \paren {T_1 \cup T_2}$ Definition of Relative Complement $\displaystyle$ $=$ $\displaystyle \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$ De Morgan's Laws: Difference with Union $\displaystyle$ $=$ $\displaystyle \relcomp S {T_1} \cap \relcomp S {T_2}$ Definition of Relative Complement

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.