# Symmetric relation

Transitive binary relations vte :
| | Symmetric | Antisymmetric | Connected | Well-founded | Has joins | Has meets | Reflexive | Irreflexive | Asymmetric |
| | | | Total, Semiconnex | | | | | Anti- reflexive | |
| Equivalence relation | Y | ✗ | ✗ | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Preorder (Quasiorder) | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Partial order | ✗ | Y | ✗ | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Total preorder | ✗ | ✗ | Y | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Total order | ✗ | Y | Y | ✗ | ✗ | ✗ | Y | ✗ | ✗ |
| Prewellordering | ✗ | ✗ | Y | Y | ✗ | ✗ | Y | ✗ | ✗ |
| Well-quasi-ordering | ✗ | ✗ | ✗ | Y | ✗ | ✗ | Y | ✗ | ✗ |
| Well-ordering | ✗ | Y | Y | Y | ✗ | ✗ | Y | ✗ | ✗ |
| Lattice | ✗ | Y | ✗ | ✗ | Y | Y | Y | ✗ | ✗ |
| Join-semilattice | ✗ | Y | ✗ | ✗ | Y | ✗ | Y | ✗ | ✗ |
| Meet-semilattice | ✗ | Y | ✗ | ✗ | ✗ | Y | Y | ✗ | ✗ |
| Strict partial order | ✗ | Y | ✗ | ✗ | ✗ | ✗ | ✗ | Y | Y |
| Strict weak order | ✗ | Y | ✗ | ✗ | ✗ | ✗ | ✗ | Y | Y |
| Strict total order | ✗ | Y | Y | ✗ | ✗ | ✗ | ✗ | Y | Y |
| | Symmetric | Antisymmetric | Connected | Well-founded | Has joins | Has meets | Reflexive | Irreflexive | Asymmetric |
| Definitions, for all $a,b$ and $S\neq \varnothing :$ | ${\begin{aligned}&aRb\\\Rightarrow&bRa\end{aligned}}$ | ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a=&b\end{aligned}}$ | ${\begin{aligned}a\neq&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ | ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ | ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ | ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ | $aRa$ | ${\text{not }}aRa$ | ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$ |
Y indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$
A term's definition may require additional properties that are not listed in this table. :
A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if:
    $\forall a,b\in X(aRb\Leftrightarrow bRa),$
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 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 a co-worker of"
* "is a 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.
Mathematical examples :
| | Symmetric | Not symmetric |
| Antisymmetric | equality | divides, less than or equal to |
| Not antisymmetric | congruence in modular arithmetic | // (integer division), most nontrivial permutations |
Non-mathematical examples :
| | 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.
* 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, 2$^{n(n+1)/2}$.
Number of n-element binary relations of different types :
| Elem­ents | Any | Transitive | Reflexive | Symmetric | 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$^{n2}$ | | 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.