# Asymmetric Relation is Antireflexive

## Theorem

Let $S$ be a set.

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

Let $\RR$ be asymmetric.

Then $\RR$ is also antireflexive.

## Proof

Let $\RR$ be asymmetric.

Then, by definition:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$

Aiming for a contradiction, suppose $\tuple {x, x} \in \RR$.

Then:

 $\displaystyle \tuple {x, x} \in \RR$ $\implies$ $\displaystyle \tuple {x, x} \notin \RR$ Definition of Asymmetric Relation $\displaystyle \leadsto \ \$ $\displaystyle \tuple {x, x} \notin \RR$  $\displaystyle$ Proof by Contradiction

Thus $\RR$ is antireflexive.

$\blacksquare$