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All definitions tacitly require the homogeneous relation be transitive: A "✓" indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. Listed here are additional properties that a homogeneous relation may satisfy. |

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:

^{[1]}

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

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.^{[1]}

## 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

By definition, a nonempty 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" |

## Properties

- A symmetric and transitive relation is always quasireflexive.
^{[citation needed]}

- A symmetric, transitive, and reflexive relation is called an equivalence relation.
^{[1]}

- 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.
^{[citation needed]}

## References

- ^
^{a}^{b}^{c}Biggs, Norman L. (2002).*Discrete Mathematics*. Oxford University Press. p. 57. ISBN 978-0-19-871369-2.

## See also

- Commutative property – Property allowing changing the order of the operands of an operation
- Symmetry in mathematics – Symmetry in mathematics
- Symmetry – Mathematical invariance under transformations