Set Difference as Symmetric Difference with Intersection

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Theorem

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

where:

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


Proof

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

$\blacksquare$