# Definition:Symmetry (Relation)

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## Contents

## 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.