Results Concerning Set Difference with Union

Theorem

 * 1) $$R \setminus \left({S \cup T}\right) = \left({R \cup T}\right) \setminus \left({S \cup T}\right)$$
 * 2) $$R \setminus \left({S \cup T}\right) = \left({R \setminus S}\right) \setminus T = \left({R \setminus T}\right) \setminus S$$
 * 3) $$\left({R \cup S}\right) \setminus T = \left({R \setminus T}\right) \cup \left({S \setminus T}\right)$$: Set difference is right distributive over union;
 * 4) $$R \setminus \left({S \setminus T}\right) = \left({R \setminus S}\right) \cup \left({R \cap T}\right)$$
 * 5) $$R \setminus S \subseteq \left({R \setminus T}\right) \cup \left({T \setminus S}\right)$$

Proof of First Assertion
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Proof of Second Assertion
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Thus $$R \setminus \left({S \cup T}\right) = \left({R \setminus S}\right) \setminus T$$.

Then:

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Proof of Third Assertion
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Proof of Fourth Assertion
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Proof of Fifth Assertion
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