Definition:Antisymmetric Relation

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Definition

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


Definition 1

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

$\tuple {x, y} \in \mathcal R \land \tuple {y, x} \in \mathcal R \implies x = y$

that is:

$\set {\tuple {x, y}, \tuple {y, x} } \subseteq \mathcal R \implies x = y$


Definition 2

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

$\tuple {x, y} \in \mathcal R \land x \ne y \implies \tuple {y, x} \notin \mathcal R$


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.