Editing Symmetric relation (section)

{{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"}}