# Restriction of Antisymmetric Relation is Antisymmetric

## Theorem

Let $S$ be a set.

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

## Proof

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

Then:

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

Thus $\mathcal R {\restriction_T}$ is antisymmetric on $T$.

$\blacksquare$