Restriction of Serial Relation is Not Necessarily Serial

Theorem
Let $S$ be a set.

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

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

Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R \restriction_T$ is not necessarily a serial relation on $T$.

Proof
Proof by Counterexample:

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

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

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

Now let $T = \set a$.

Then $\mathcal R {\restriction_T} = \O$.

So $\not \exists y \in T: \tuple {a, y} \in \mathcal R {\restriction_T}$.

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

Also see

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