# Restriction of Antisymmetric Relation is Antisymmetric

## Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be an antisymmetric 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 an antisymmetric relation on $T$.

## Proof

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

Then:

 $\displaystyle \set {\tuple {x, y}, \tuple {y, x} }$ $\subseteq$ $\displaystyle \RR {\restriction_T}$ $\displaystyle \leadsto \ \$ $\displaystyle \set {\tuple {x, y}, \tuple {y, x} }$ $\subseteq$ $\displaystyle \paren {T \times T} \cap \RR$ Definition of Restriction of Relation $\displaystyle \leadsto \ \$ $\displaystyle \set {\tuple {x, y}, \tuple {y, x} }$ $\subseteq$ $\displaystyle \RR$ Intersection is Subset $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y$ $\RR$ is Antisymmetric on $S$

Thus $\RR {\restriction_T}$ is antisymmetric on $T$.

$\blacksquare$