## Frequent Links

# Symmetric relation

In mathematics and other areas, a binary relation *R* over a set *X* is **symmetric** if it holds for all *a* and *b* in *X* that if *a* is related to *b* then *b* is related to *a*.

In mathematical notation, this is:

- <math>\forall a, b \in X,\ a R b \Rightarrow \; b R a.</math>

## Contents

## Examples

### In mathematics

- "is equal to" (equality) (whereas "is less than" is not symmetric)
- "is comparable to", for elements of a partially ordered set
- "... and ... are odd":

### Outside mathematics

- "is married to" (in most legal systems)
- "is a fully biological sibling of"
- "is a homophone of"

## Relationship to asymmetric and antisymmetric relations

By definition, a relation cannot be both symmetric and asymmetric (where if *a* is related to *b*, then *b* cannot be related to *a* (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").

Symmetric and antisymmetric (where the only way *a* can be related to *b* and *b* be related to *a* is if *a* = *b*) are actually independent of each other, as these examples show.

Symmetric |
Not symmetric
| |

Antisymmetric |
equality | "is less than or equal to" |

Not antisymmetric |
congruence in modular arithmetic | "is divisible by", over the set of integers |

Symmetric |
Not symmetric
| |

Antisymmetric |
"is the same person as, and is married" | "is the plural of" |

Not antisymmetric |
"is a full biological sibling of" | "preys on" |

## Additional aspects

A symmetric relation that is also transitive and reflexive is an equivalence relation.

One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge's two vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatorially equivalent objects.