# 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 $\mathcal R$ defined by:

$\mathcal R = \displaystyle \bigcap \mathscr R$

where $\bigcap$ denotes set intersection.

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

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