Definition:Intersection of Relations

Definition

Let $S$ and $T$ be sets.

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

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

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

where $\cap$ denotes set intersection.

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

$s \mathrel{\mathcal Q} t$ iff both $s \mathrel{\mathcal R_1} t$ and $s \mathrel{\mathcal R_2} t$

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 $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$