Set Difference Union Intersection/Proof 2

Theorem

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


Proof

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

$\blacksquare$

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