View source for Symmetric relation

{{Short description|Type of binary relation}}{{stack|{{Binary relations}}}}

A '''symmetric relation''' is a type of [[binary relation]]. Formally, a binary relation ''R'' over a [[Set (mathematics)|set]] ''X'' is symmetric if:{{refn|name=":0"|{{Cite book|last=Biggs|first=Norman L.|title=Discrete Mathematics|publisher=Oxford University Press|year=2002|isbn=978-0-19-871369-2|page=57}}}}
: <math>\forall a, b \in X(a R b \Leftrightarrow b R a) ,</math>
where the notation ''aRb'' means that {{nowrap|(''a'', ''b'') ∈ ''R''}}.

An example is the relation "is equal to", because if {{nowrap|1=''a'' = ''b''}} is true then {{nowrap|1=''b'' = ''a''}} is also true. If ''R''<sup>T</sup> represents the [[converse relation|converse]] of ''R'', then ''R'' is symmetric if and only if {{nowrap|1=''R'' = ''R''<sup>T</sup>}}.<ref name="Characterization of Symmetric Relations">{{cite web |title=MAD3105 1.2 |url=https://www.math.fsu.edu/~pkirby/mad3105/index.math.htm#:~:text=%C2%A0%20Course%20Notes%3A%201.2%20Closure%20of%20Relations |website=Florida State University Department of Mathematics |publisher=Florida State University |access-date=30 March 2024}}</ref>

Symmetry, along with [[Reflexive relation|reflexivity]] and [[Transitive relation|transitivity]], are the three defining properties of an [[equivalence relation]].{{refn|name=":0"}}

== Examples ==
=== In mathematics ===
* "is equal to" ([[equality (mathematics)|equality]]) (whereas "is less than" is not symmetric)
* "is [[comparability|comparable]] to", for elements of a [[partially ordered set]]
* "... and ... are odd":
::::::[[Image:Bothodd.png]]

=== 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 ==

[[File:Symmetric-and-or-antisymmetric.svg|thumb|Symmetric and antisymmetric relations|340x340px]]

By definition, a nonempty relation cannot be both symmetric and [[asymmetric relation|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 relation|antisymmetric]] (where the only way ''a'' can be related to ''b'' and ''b'' be related to ''a'' is if {{nowrap|1=''a'' = ''b''}}) are actually independent of each other, as these examples show.

{| class="wikitable"
|+Mathematical examples
|-
| || '''Symmetric''' || '''Not symmetric'''
|-
| '''Antisymmetric''' || [[equality (mathematics)|equality]] || [[Divisor|divides]], less than or equal to
|-
| '''Not antisymmetric''' || [[congruence relation|congruence]] in [[modular arithmetic]] || [[integer division|//]] (integer division), most nontrivial [[Permutation|permutations]]
|}

{| class="wikitable"
|+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 relation|quasireflexive]].{{efn|If ''xRy'', the ''yRx'' by symmetry, hence ''xRx'' by transitivity. The proof of {{nowrap|''xRy'' ⇒ ''yRy''}} is similar.}}
* 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 {{nowrap|''n'' × ''n''}} binary upper triangle matrices, 2<sup>''n''(''n''+1)/2</sup>.{{refn|{{Cite OEIS|A006125}}}}

{{Number of relations}}

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== See also ==
* {{annotated link|Commutative property}}
* {{annotated link|Symmetry in mathematics}}
* {{annotated link|Symmetry}}

[[Category:Properties of binary relations]]
[[Category:Symmetric relations| ]]