Results Concerning Set Difference with Intersection

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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

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


Set Intersection Distributes over Set Difference

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


Also see

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

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