Set Difference with Intersection

Theorem

 * The set difference with the intersection is just the set difference:
 * $$S - \left({S \cap T}\right) = S - T$$


 * The set difference of the intersection of two sets with one of those sets is the empty set:
 * $$\left({S \cap T}\right) - S = \varnothing$$


 * The intersection of the set difference with the first set is the set difference:
 * $$\left({S - T}\right) \cap S = S - T$$


 * The intersection of the set difference with the second set is the empty set:
 * $$\left({S - T}\right) \cap T = \varnothing$$

Proof of First Assertion
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Proof of Second Assertion
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Proof of Third Assertion
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Proof of Fourth Assertion
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