Definition:Antisymmetric Relation

Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

Definition 1

$\mathcal R$ is antisymmetric if and only if:

$\left({x, y}\right) \in \mathcal R \land \left({y, x}\right) \in \mathcal R \implies x = y$

that is:

$\left\{{\left({x, y}\right), \left({y, x}\right)}\right\} \subseteq \mathcal R \implies x = y$

Definition 2

$\mathcal R$ is antisymmetric if and only if:

$\left({x, y}\right) \in \mathcal R \land x \ne y \implies \left({y, x}\right) \notin \mathcal R$

Also known as

Some sources render this concept as anti-symmetric relation.

Also see

Note the difference between:

An asymmetric relation, in which the fact that $\left({x, y}\right) \in \mathcal R$ means that $\left({y, x}\right)$ is definitely not in $\mathcal R$

and:

An antisymmetric relation, in which there may be instances of both $\left({x, y}\right) \in \mathcal R$ and $\left({y, x}\right) \in \mathcal R$ but if there are, then it means that $x$ and $y$ have to be the same object.

• Results about symmetry of relations can be found here.