Definition:Intersection of Relations
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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 $s \mathrel{\mathcal R_2} t$
General Definition
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 $\mathcal Q \in \mathscr R$, $s \mathrel{\mathcal Q} t$