Union with Set Difference

Theorem

 * The union of a set difference with the second set is the union of the two sets:
 * $$\left({S \setminus T}\right) \cup T = S \cup T$$


 * The union of a set difference with the first set is the set itself:
 * $$\left({S \setminus T}\right) \cup S = S$$


 * The set difference between two sets is the same as the set difference between their union and the second of the two sets:
 * $$\left({S \cup T}\right) \setminus T = S \setminus T$$

Proof of First Assertion
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Proof of Second Assertion
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Proof of Third Assertion
From Results Concerning Set Difference with Union:
 * $$\left({R \cup S}\right) \setminus T = \left({R \setminus T}\right) \cup \left({S \setminus T}\right)$$

Hence:

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