Category:Definitions/Symmetry (Relations)

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This category contains definitions related to Symmetry (Relations).


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.