Properties of Restriction of Relation

Theorem
Let $$\left({S, \mathcal R}\right)$$ be a relational structure.

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

Let $$\left({T, \mathcal R \restriction_T}\right)$$ 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

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

Reflexivity

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

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

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


 * 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 $$T$$.

Symmetry

 * 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.


 * 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.


 * Now suppose $$\mathcal R$$ is antisymmetric.

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

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

Transitivity

 * Suppose $$\mathcal R$$ is transitive.

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

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


 * Suppose $$\mathcal R$$ is antitransitive.

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

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

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

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