Set Difference Intersection with Second Set is Empty Set/Proof 1

Theorem

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

Proof

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

\(=\)

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

Set Intersection Distributes over Set Difference

\(\ds \)

\(=\)

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

Set Intersection is Idempotent

\(\ds \)

\(=\)

\(\ds \O\)

Set Difference of Intersection with Set is Empty Set

$\blacksquare$