# Symmetric relation

A **symmetric relation** is a type of binary relation. An example is the relation "is equal to", because if *a* = *b* is true then *b* = *a* is also true. Formally, a binary relation *R* over a set *X* is symmetric if and only if:

If *R*^{T} represents the converse of *R*, then *R* is symmetric if and only if *R* = *R*^{T}.

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

## 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"
- "is co-worker of"
- "is teammate of"

## Relationship to asymmetric and antisymmetric relations

A nonempty relation can be both symmetric and asymmetric (consider the relation of equality). 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.