Union Distributes over Intersection/Venn Diagram

Theorem

Set union is distributive over set intersection:

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

Proof

Demonstration by Venn diagram:

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

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

Their union $R \cup \paren {S \cap T}$ is the total shaded area: yellow, blue and green.

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

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

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.2$
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Ponderable $1.2.1 \ \text{(b)}$