# Intersection with Set Difference is Set Difference with Intersection/Proof 1

## Theorem

$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$

## Proof

Consider $R, S, T \subseteq \mathbb U$, where $\mathbb U$ is considered as the universe.

Then:

 $\displaystyle \left({R \setminus S}\right) \cap T$ $=$ $\displaystyle \left({R \cap \complement \left({S}\right)}\right) \cap T$ Set Difference as Intersection with Complement $\displaystyle$ $=$ $\displaystyle \left({R \cap T}\right) \cap \complement \left({S}\right)$ Intersection is Commutative and Intersection is Associative $\displaystyle$ $=$ $\displaystyle \left({R \cap T}\right) \setminus S$ Set Difference as Intersection with Complement

$\blacksquare$