Results Concerning Set Difference with Intersection

Theorem
Various results concerning set difference and set intersection:


 * 1) $$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$$;
 * 2) $$\left({R \cap S}\right) \setminus T = \left({R \setminus T}\right) \cap \left({S \setminus T}\right)$$: Set difference is right distributive over set intersection.
 * 3) $$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus \left({S \cap T}\right)$$: Set intersection is right distributive over set difference;
 * 4) $$R \cap \left({S \setminus T}\right) = \left({R \cap S}\right) \setminus \left({R \cap T}\right)$$: Set intersection is left distributive over set difference;

Proof of First Assertion
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Proof of Second Assertion
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Proof of Third Assertion
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Note that this proof requires the use of $$\left({S \cap T}\right) \setminus S = \varnothing$$, which follows from the second assertion above.

Proof of Fourth Assertion
Follows directly from the third assertion above:

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