Set Difference as Symmetric Difference with Intersection

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Theorem

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

where:

$S \setminus T$ denotes set difference
$S * T$ denotes set symmetric difference
$S \cap T$ denotes set intersection.


Proof

\(\displaystyle S * \left({S \cap T}\right)\) \(=\) \(\displaystyle \left({S \setminus \left({S \cap T}\right)}\right) \cup \left({\left({S \cap T}\right) \setminus S}\right)\) Definition of Symmetric Difference
\(\displaystyle \) \(=\) \(\displaystyle \left({S \setminus \left({S \cap T}\right)}\right) \cup \varnothing\) Set Difference of Intersection with Set is Empty Set
\(\displaystyle \) \(=\) \(\displaystyle \left({S \setminus T}\right) \cup \varnothing\) Set Difference with Intersection is Difference
\(\displaystyle \) \(=\) \(\displaystyle S \setminus T\) Union with Empty Set

$\blacksquare$