Properties of Restriction of Relation

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a 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 $T$.

If $\mathcal R$ on $S$ has any of the properties:


 * Reflexive
 * Antireflexive
 * Symmetric
 * Antisymmetric
 * Asymmetric
 * Transitive
 * Antitransitive
 * Connected

... then $\mathcal R \restriction_T$ on $T$ has the same properties.

Restriction of Connected Relation is Connected

 * Suppose $\mathcal R$ is connected.

Let $x, y \in T$.

As $T \subseteq S$, from the definition of subset, $x, y \in S$.

As $\mathcal R$ is by definition a connected relation, either $\left({x, y}\right) \in \mathcal R$ or $\left({y, x}\right) \in \mathcal R$.

As $x$ and $y$ are arbitrary elements of $T$ it follows that $\left({T, \preceq}\right)$ is also connected.

Note
If a relation is:
 * serial,
 * non-reflexive,
 * non-symmetric,
 * non-transitive or
 * non-connected

it is impossible to state without further information whether or not any restriction of that relation has the same properties.