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