Results Concerning Set Difference with Union

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Theorem

Let:

$S \setminus T$ denote set difference
$S \cup T$ denote set union
$S \cap T$ denote set intersection.


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

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


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

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