# Definition:Union of Relations

## Definition

Let $S$ and $T$ be sets.

Let $\mathcal R_1$ and $\mathcal R_2$ be relations on $S \times T$.

The union of $\mathcal R_1$ and $\mathcal R_2$ is the relation $\mathcal Q$ defined by:

$\mathcal Q := \mathcal R_1 \cup \mathcal R_2$

where $\cup$ denotes set union.

Explicitly, for $s \in S$ and $t \in T$, we have:

$s \mathrel{\mathcal Q} t$ if and only if $s \mathrel{\mathcal R_1} t$ or $s \mathrel{\mathcal R_2} t$

### General Definition

Let $\mathscr R$ be a collection of relations on $S \times T$.

The union of $\mathscr R$ is the relation $\mathcal R$ defined by:

$\mathcal R = \displaystyle \bigcup \mathscr R$

where $\bigcup$ denotes set union.

Explicitly, for $s \in S$ and $t \in T$:

$s \mathrel{\mathcal R} t$ if and only if for some $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$