# Absorption Laws (Set Theory)/Intersection with Union

## Theorem

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

## 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

$\blacksquare$

## 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

$\blacksquare$

## Also see

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