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

$\blacksquare$