Restriction of Asymmetric Relation is Asymmetric

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a asymmetric relation on $S$.

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

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is a asymmetric relation on $T$.

Suppose $\RR$ is asymmetric on $S$.

Then:

$\ds \tuple {x, y}$

$\in$

$\ds \RR {\restriction_T}$

$\ds \leadsto \ \ $

$\ds \tuple {x, y}$

$\in$

$\ds \paren {T \times T} \cap \RR$

$\ds \leadsto \ \ $

$\ds \tuple {x, y}$

$\in$

$\ds \RR$

$\ds \leadsto \ \ $

$\ds \tuple {y, x}$

$\notin$

$\ds \RR$

$\RR$ is asymmetric on $S$

$\ds \leadsto \ \ $

$\ds \tuple {y, x}$

$\notin$

$\ds \RR {\restriction_T}$

and so $\RR {\restriction_T}$ is asymmetric on $T$.

$\blacksquare$