Definition:Symmetry (Relation)

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Definition

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


Symmetric

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

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


Asymmetric

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

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


Antisymmetric

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

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

that is:

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


Non-symmetric

$\mathcal R$ is non-symmetric if and only if it is neither symmetric nor asymmetric.


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.


Linguistic Note

The word symmetry comes from Greek συμμετρεῖν (symmetría) meaning measure together.


Also see

  • Results about symmetry of relations can be found here.