Definition:Antisymmetric Relation/Definition 1

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Definition

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

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

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

that is:

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


Also known as

  • Some sources render this concept as anti-symmetric relation.


Antisymmetric and Asymmetric Relations

Note the difference between:

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

and:

An antisymmetric relation, in which there may be instances of both $\tuple {x, y} \in \mathcal R$ and $\tuple {y, x} \in \mathcal R$ 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.


Sources