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:

\forall a,b\in X(aRb\Leftrightarrow bRa),

[1]

Transitive binary relations

	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 ${\displaystyle S\neq \varnothing$	${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$	${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$	${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$	${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$	${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$	$aRa$	${\text{not }}aRa$	${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$

indicates that the column's property is always true 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

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.

where the notation $aRb$ means that $(a,b)\in R$ .

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.[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

Symmetric 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.[2]

A symmetric, transitive, and reflexive relation is called an equivalence relation.[1]

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 nxn binary upper triangle matrices, $2^{n(n+1)/2}.$ [3]

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		2ⁿ⁽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.

References

Biggs, Norman L. (2002). Discrete Mathematics. Oxford University Press. p. 57. ISBN 978-0-19-871369-2.
If xRy, the yRx by symmetry, hence xRx by transitivity. The proof of xRy ⇒ yRy is similar.
Sloane, N. J. A. (ed.). "Sequence A006125". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.