# De Morgan's Laws (Set Theory)/Set Difference/Difference with Union/Venn Diagram

## Theorem

$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$

## Proof

Demonstration by Venn diagram:

The area in orange and red is the set difference of $S$ with $T_1$

The area in orange and yellow is set difference of $S$ with $T_2$

The area in orange only is the set difference of $S$ with the union of $T_1$ and $T_2$.

It is also seen to be the intersection of the set difference of $S$ with $T_1$ and the set difference of $S$ with $T_2$.

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.