# Union Distributes over Intersection/Proof 1

## Theorem

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

## Proof

 $\ds$  $\ds x \in R \cup \paren {S \cap T}$ $\ds$ $\leadstoandfrom$ $\ds x \in R \lor \paren {x \in S \land x \in T}$ Definition of Set Union and Definition of Set Intersection $\ds$ $\leadstoandfrom$ $\ds \paren {x \in R \lor x \in S} \land \paren {x \in R \lor x \in T}$ Disjunction is Left Distributive over Conjunction $\ds$ $\leadstoandfrom$ $\ds x \in \paren {R \cup S} \cap \paren {R \cup T}$ Definition of Set Union and Definition of Set Intersection

$\blacksquare$