Restriction of Serial Relation is Not Necessarily Serial

Theorem
Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a serial relation on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is not necessarily a serial relation on $T$.

Proof
Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, b}, \tuple {b, b} }$.

$\RR$ is a serial relation, as can be seen by definition.

Now let $T = \set a$.

Then:
 * $\RR {\restriction_T} = \O$

So:
 * $\not \exists y \in T: \tuple {a, y} \in \RR {\restriction_T}$

That is, $\RR {\restriction_T}$ is not a serial relation on $T$.

Also see

 * Properties of Relation Not Preserved by Restriction‎ for other similar results.