Results Concerning Set Difference with Union

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Theorem

Let:


Set Difference with Union

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


Set Difference is Right Distributive over Union

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


Set Difference with Set Difference is Union of Set Difference with Intersection

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


Set Difference is Subset of Union of Differences

$R \setminus S \subseteq \left({R \setminus T}\right) \cup \left({T \setminus S}\right)$