Set Difference with Intersection

Theorem

Let $S$ and $T$ be sets.

Set Difference with Intersection is Difference

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

Set Difference of Intersection with Set is Empty Set

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

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

Set Difference Intersection with First Set is Set Difference

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

Set Difference Intersection with Second Set is Empty Set

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

Set Difference with Set Difference

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

Also see

• De Morgan's Laws: Difference with Intersection