# Definition:Symmetry (Relation)

## 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.