Definition:Intersection of Relations/General Definition

Definition

Let $S$ and $T$ be sets.

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

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

$\ds \RR = \bigcap \mathscr R$

where $\bigcap$ denotes set intersection.

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

$s \mathrel \RR t$ if and only if:

$\forall \QQ \in \mathscr R: s \mathrel \QQ t$