# Set Difference with Intersection is Difference

## Theorem

The set difference with the intersection is just the set difference.

Let $S, T$ be sets.

Then:

$S \setminus \paren {S \cap T} = S \setminus T$

## Proof

 $\ds S \setminus \paren {S \cap T}$ $=$ $\ds \paren {S \setminus S} \cup \paren {S \setminus T}$ De Morgan's Laws: Difference with Intersection $\ds$ $=$ $\ds \O \cup \paren {S \setminus T}$ Set Difference with Self is Empty Set $\ds$ $=$ $\ds S \setminus T$ Union with Empty Set

$\blacksquare$