Definition:Intersection of Relations

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Definition

Let $S$ and $T$ be sets.

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


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

$\QQ := \RR_1 \cap \RR_2$

where $\cap$ denotes set intersection.


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

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


General Definition

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


Also see