Set Difference as Symmetric Difference with Intersection

Theorem

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

where:

$S \setminus T$ denotes set difference

$S \symdif T$ denotes set symmetric difference

$S \cap T$ denotes set intersection.

Proof

\(\ds S \symdif \paren {S \cap T}\)

\(=\)

\(\ds \paren {S \setminus \paren {S \cap T} } \cup \paren {\paren {S \cap T} \setminus S}\)

Definition of Symmetric Difference

\(\ds \)

\(=\)

\(\ds \paren {S \setminus \paren {S \cap T} } \cup \O\)

Set Difference of Intersection with Set is Empty Set

\(\ds \)

\(=\)

\(\ds \paren {S \setminus T} \cup \O\)

Set Difference with Intersection is Difference

\(\ds \)

\(=\)

\(\ds S \setminus T\)

Union with Empty Set

$\blacksquare$