Properties of Restriction of Relation

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

Let $$\left({T; \mathcal{R}|_T}\right)$$ be the restriction of $$\mathcal{R}$$ to $$T$$.

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


 * Reflexive
 * Symmetric
 * Transitive
 * Antitransitive

... then $$\mathcal{R}|_T$$ on $$T$$ has the same properties.