Restriction of Asymmetric Relation is Asymmetric

Theorem

Let $S$ be a set.

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

Proof

Suppose $\mathcal R$ is asymmetric on $S$.

Then:

 $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R \restriction_T$ $\displaystyle \implies \ \$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \left({T \times T}\right) \cap \mathcal R$ Definition of Restriction of Relation $\displaystyle \implies \ \$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R$ Intersection is Subset $\displaystyle \implies \ \$ $\displaystyle \left({y, x}\right)$ $\notin$ $\displaystyle \mathcal R$ $\mathcal R$ is asymmetric on $S$ $\displaystyle \implies \ \$ $\displaystyle \left({y, x}\right)$ $\notin$ $\displaystyle \mathcal R \restriction_T$ Definition of Restriction of Relation

and so $\mathcal R \restriction_T$ is asymmetric on $T$.

$\blacksquare$