Absorption Laws (Set Theory)/Intersection with Union

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

Proof 1

\(\ds \) \(\) \(\ds S \subseteq \paren {S \cup T}\) Set is Subset of Union
\(\ds \) \(\leadsto\) \(\ds S \cap \paren {S \cup T} = S\) Intersection with Subset is Subset‎


Proof 2

\(\ds x\) \(\in\) \(\ds S \cap \paren {S \cup T}\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds S \land \paren {x \in S \lor x \in T}\) Definition of Set Intersection and Definition of Set Union
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds S\) Conjunction Absorbs Disjunction


Also see

These two results together are known as the Absorption Laws, corresponding to the equivalent results in logic.