# Set Difference with Set Difference

## Theorem

The set difference with the set difference of two sets is the intersection of the two sets:

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

## Proof

 $\displaystyle S \setminus \paren {S \setminus T}$ $=$ $\displaystyle \paren {S \setminus S} \cup \paren {S \cap T}$ Set Difference with Set Difference is Union of Set Difference with Intersection $\displaystyle$ $=$ $\displaystyle \O \cup \paren {S \cap T}$ Set Difference with Self is Empty Set $\displaystyle$ $=$ $\displaystyle S \cap T$ Union with Empty Set

Interchanging $S$ and $T$:

 $\displaystyle T \setminus \paren {T \setminus S}$ $=$ $\displaystyle T \cap S$ $\displaystyle$ $=$ $\displaystyle S \cap T$ Intersection is Commutative

$\blacksquare$