Definition:Union of Relations

Definition

Let $S$ and $T$ be sets.

Let $\RR_1$ and $\RR_2$ be relations on $S \times T$.

The union of $\RR_1$ and $\RR_2$ is the relation $\QQ$ defined by:

$\QQ := \RR_1 \cup \RR_2$

where $\cup$ denotes set union.

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

$s \mathrel \QQ t$ if and only if $s \mathrel {\RR_1} t$ or $s \mathrel {\RR_2} t$

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

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

$\ds \RR = \bigcup \mathscr R$

where $\bigcup$ denotes set union.

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

$s \mathrel \RR t$ if and only if for some $\QQ \in \mathscr R$, $s \mathrel \QQ t$