# Union Distributes over Intersection

## Theorem

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

### Family of Sets

Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of a set $S$.

Let $B \subseteq S$.

Then:

$\ds \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

where $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection of $\family {A_\alpha}_{\alpha \mathop \in I}$.

### General Result

Let $S$ and $T$ be sets.

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T$ be a subset of $\powerset T$.

Then:

$\ds S \cup \bigcap \mathbb T = \bigcap_{X \mathop \in \mathbb T} \paren {S \cup X}$

## Proof 1

 $\ds$  $\ds x \in R \cup \paren {S \cap T}$ $\ds$ $\leadstoandfrom$ $\ds x \in R \lor \paren {x \in S \land x \in T}$ 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

$\blacksquare$

## Proof 2

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

From the Duality Principle for Sets, exchanging $\cup$ for $\cap$ throughout, and vice versa, reveals the result:

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

$\blacksquare$

## 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.

## Also known as

This result and Intersection Distributes over Union are together known as the Distributive Laws.

## Examples

### Arbitrary Integer Sets: $1$

$A \cup \paren {B \cap C} = \set {1, 2, 3, 4, 6, 8, \dotsc} = \paren {A \cup B} \cap \paren {A \cup C}$

### Arbitrary Integer Sets: $2$

$B \cup \paren {A \cap C} = \set {1, 2, 3, 4, 5, 7, \dotsc} = \paren {B \cup A} \cap \paren {B \cup C}$