# Set Difference Intersection with Second Set is Empty Set

## Theorem

Let $S$ and $T$ be sets.

The intersection of the set difference of $S$ and $T$ with $T$ is the empty set:

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

## Proof 1

 $\ds \paren {S \setminus T} \cap T$ $=$ $\ds \paren {S \cap T} \setminus \paren {T \cap T}$ Set Intersection Distributes over Set Difference $\ds$ $=$ $\ds \paren {S \cap T} \setminus T$ Set Intersection is Idempotent $\ds$ $=$ $\ds \O$ Set Difference of Intersection with Set is Empty Set

$\blacksquare$

## Proof 2

 $\ds \paren {S \setminus T} \cap T$ $=$ $\ds \paren {S \cap T} \setminus T$ Intersection with Set Difference is Set Difference with Intersection $\ds$ $=$ $\ds \O$ Set Difference of Intersection with Set is Empty Set

$\blacksquare$