Set Difference Disjoint with Reverse

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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

$\blacksquare$