Definition:Intersection of Relations
Jump to navigation
Jump to search
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:
- $\forall \QQ \in \mathscr R: s \mathrel \QQ t$