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

## 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:

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

## Proof

Let $T_1, T_2 \subseteq S$.

Then from Intersection is Subset and Subset Relation is Transitive:

$T_1 \cap T_2 \subseteq S$

Hence:

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

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.