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A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if:cite-ref-0-1-0[1]

∀ ∀ a , b ∈ ∈ X ( a R b ⇔ ⇔ b R a ) , {\displaystyle \forall a,b\in X(aRb\Leftrightarrow bRa),}

where the notation aRb means that (a, b) ∈ R.

An example is the relation "is equal to", because if a = b is true then b = a is also true. If RT represents the converse of R, then R is symmetric if and only if R = RT.cite-ref-characterization-of-symmetric-relations-2-0[2]

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.cite-ref-0-1-1[1]

• Examples

• In mathematics

• Outside mathematics

• Relationship to asymmetric and antisymmetric relations

• Properties

• Notes

• References

• See also

• "is equal to" (equality) (whereas "is less than" is not symmetric)

• "is comparable to", for elements of a partially ordered set

• "... and ... are odd":

• "is married to" (in most legal systems)

• "is a fully biological sibling of"

• "is a homophone of"

• "is a co-worker of"

• "is a teammate of"

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 | divides , less than or equal to |

| Not antisymmetric | congruence in modular arithmetic | // (integer division), most nontrivial permutations |

| | 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 |

• A symmetric and transitive relation is always quasireflexive.cite-ref-3[a]

• One way to count the symmetric relations on n elements, that in their binary matrix representation the upper right triangle determines the relation fully, and it can be arbitrary given, thus there are as many symmetric relations as n × n binary upper triangle matrices, 2n(n+1)/2.cite-ref-4[3]

| Elem­ents | Any | Transitive | Reflexive | | Preorder | Partial order | Total preorder | Total order | Equivalence relation |

|---|---|---|---|---|---|---|---|---|---|

| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

| 1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |

| 2 | 16 | 13 | 4 | 8 | 4 | 3 | 3 | 2 | 2 |

| 3 | 512 | 171 | 64 | 64 | 29 | 19 | 13 | 6 | 5 |

| 4 | 65,536 | 3,994 | 4,096 | 1,024 | 355 | 219 | 75 | 24 | 15 |

| n | 2 n 2 | | 2 n ( n −1) | 2 n ( n +1)/2 | | | ∑ n k =0 k ! S ( n , k ) | n ! | ∑ n k =0 S ( n , k ) |

| OEIS | A002416 | A006905 | A053763 | A006125 | A000798 | A001035 | A000670 | A000142 | A000110 |

Note that S(n, k) refers to Stirling numbers of the second kind.

cite-note-3a. ↑ If xRy, the yRx by symmetry, hence xRx by transitivity. The proof of xRy ⇒ yRy is similar.

cite-note-0-11. ↑ citerefbiggs2002Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 978-0-19-871369-2.

cite-note-characterization-of-symmetric-relations-22. ↑ "MAD3105 1.2". Florida State University Department of Mathematics. Florida State University. Retrieved 30 March 2024.

cite-note-43. ↑ citerefsloane-a006125Sloane, N. J. A. (ed.). "Sequence A006125". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

• Commutative property – Property of some mathematical operations

• Symmetry in mathematics

• Symmetry – Mathematical invariance under transformations