Absorption Laws (Set Theory)/Union with Intersection

Theorem

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

Proof 1

 $\displaystyle$  $\displaystyle \paren {S \cap T} \subseteq S$ Intersection is Subset $\displaystyle$ $\leadsto$ $\displaystyle S \cup \paren {S \cap T} = S$ Union with Superset is Superset‎

$\blacksquare$

Proof 2

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

$\blacksquare$

Also see

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