Definition:Antisymmetric Relation/Definition 2

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

$\RR$ is antisymmetric if and only if:

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

Also known as

Some sources render this concept as anti-symmetric relation.

Some sources (perhaps erroneously) use this definition for an asymmetric relation.

Antisymmetric and Asymmetric Relations

Note the difference between:

An asymmetric relation, in which the fact that $\tuple {x, y} \in \RR$ means that $\tuple {y, x}$ is definitely not in $\RR$


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

Also see

  • Results about symmetry of relations can be found here.