Results Concerning Set Difference with Intersection

Theorem

Let:

$S \setminus T$ denote set difference

$S \cap T$ denote set intersection.

Intersection with Set Difference is Set Difference with Intersection

$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$

Set Difference is Right Distributive over Set Intersection

$\paren {A \cap B} \setminus C = \paren {A \setminus C} \cap \paren {B \setminus C}$

Set Intersection Distributes over Set Difference

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

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

Also see

• De Morgan's Laws: Difference with Intersection, in which:

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

shows that set difference is not left distributive over set intersection.