Intersection of Transitive Relations is Transitive/General Result

Theorem

Let $\family {\RR_i: i \mathop \in I}$ be an $I$-indexed set of transitive relations on a set $S$.

Then their intersection $\ds \bigcap_{i \mathop \in I} \RR_i$ is also a transitive relation on $S$.

Proof

Let $\ds \tuple {x, y} \in \bigcap_{i \mathop \in I} \RR_i$ and $\ds \tuple {y, z} \in \bigcap_{i \mathop \in I} \RR_i$ be transitive relations on an arbitrary set $S$.

Then by definition of intersection:

$\tuple {x, y} \in \RR_i$ for all $i \in I$

$\tuple {y, z} \in \RR_i$ for all $i \in I$

Since each $\RR_i$ is transitive:

$\tuple {x, z} \in \RR_i$ for all $i \in I$

so, by definition of intersection:

$\ds \tuple {x, z} \in \bigcap_{i \mathop \in I} \RR_i$

Hence $\ds \bigcap_{i \mathop \in I} \RR_i$ is transitive.

$\blacksquare$