# Category:Definitions/Symmetry (Relations)

This category contains definitions related to Symmetry (Relations).

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

### Symmetric

$\RR$ is symmetric if and only if:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

### Asymmetric

$\RR$ is asymmetric if and only if:

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

### Antisymmetric

$\RR$ is antisymmetric if and only if:

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

that is:

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

### Non-symmetric

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

## Pages in category "Definitions/Symmetry (Relations)"

The following 14 pages are in this category, out of 14 total.