# Intersection Distributes over Union/Venn Diagram

## Theorem

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

## Proof

Demonstration by Venn diagram:

In the left hand diagram, $R$ is depicted in yellow and $S \cup T$ is depicted in blue.

Their intersection $R \cap \paren {S \cup T}$ where they overlap is depicted in green.

In the right hand diagram, $\paren {R \cap S}$ is depicted in yellow and $\paren {R \cap T}$ is depicted in blue.

Their intersection, where they overlap, is depicted in green.

Their union is the total shaded area: yellow, blue and green.

As can be seen by inspection, the areas are the same.