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

## Theorem

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

## Proof

Demonstration by Venn diagram:

The area in blue and magenta is the set difference of $S$ with $T_1$

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

The complete shaded area is the set difference of $S$ with the intersection of $T_1$ and $T_2$.

It is also seen to be the union 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.