Set Difference Disjoint with Reverse

Theorem

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

Proof

We assume that $S, T \subseteq \mathbb U$ where $\mathbb U$ is the universe.

Then we can use the definition of Set Difference as Intersection with Complement.

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

$\blacksquare$