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 Antireflexive Relation is Antireflexive

 * Suppose $\mathcal R$ is antireflexive on $S$.

Then $\forall x \in S: \left({x, x}\right) \notin \mathcal R$, and so $\forall x \in T: \left({x, x}\right) \notin \mathcal R \restriction_T$.

Thus $\mathcal R \restriction_T$ is antireflexive on $T$.

Restriction of Symmetric Relation is Symmetric

 * Suppose $\mathcal R$ is symmetric on $S$.

Then $\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R$.

So, if both $x$ and $y$ are in $T$, $\left({x, y}\right) \in \mathcal R \restriction_T$ and $\left({y, x}\right) \in \mathcal R \restriction_T$ and so $\mathcal R \restriction_T$ is symmetric.

Restriction of Asymmetric Relation is Asymmetric

 * Similarly for asymmetry.

Let $\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \notin \mathcal R$.

If both $x$ and $y$ are in $T$, $\left({x, y}\right) \in \mathcal R \restriction_T$ but $\left({y, x}\right) \notin \mathcal R \restriction_T$ still.

And so $\mathcal R \restriction_T$ is asymmetric.

Restriction of Antisymmetric Relation is Antisymmetric

 * Now suppose $\mathcal R$ is antisymmetric.

Then $\left({x, y}\right) \in \mathcal R \land \left({y, x}\right) \in \mathcal R \implies x = y$.

By the above argument, the same applies to $\mathcal R \restriction_T$.

Restriction of Transitive Relation is Transitive

 * Suppose $\mathcal R$ is transitive.

Then $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Therefore, if $x, y, z \in T$, it follows that $\left({x, y}\right) \in \mathcal R \restriction_T, \left({y, z}\right) \in \mathcal R \restriction_T \implies \left({x, z}\right) \in \mathcal R \restriction_T$.

Restriction of Antitransitive Relation is Antitransitive

 * Suppose $\mathcal R$ is antitransitive.

Then there are no $\left({x, z}\right) \in \mathcal R$ such that $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R$.

If $x, y, z \in T$, then the same still applies, and $\mathcal R \restriction_T$ remains antitransitive.

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:
 * 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.