Restriction of Asymmetric Relation is Asymmetric

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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 $T$.

$\blacksquare$


Also see