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 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}}}}

: $\forall a, b \in X(a R b \Leftrightarrow b R a) ,$

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^T represents the converse of R, then R is symmetric if and only if {{nowrap|1=R = R^T}}.{{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}}

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

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":

::::::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

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 {{nowrap|1=a = b}}) are actually independent of each other, as these examples show.

class="wikitable" \|+Mathematical examples
	Symmetric	Not symmetric
Antisymmetric	equality	divides, less than or equal to
Not antisymmetric	congruence in modular arithmetic	// (integer division), most nontrivial 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.{{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^n(n+1)/2.{{refn|{{Cite OEIS|A006125}}}}

Symmetric relation

Examples

= In mathematics =

= Outside mathematics =

Relationship to asymmetric and antisymmetric relations

Properties

Notes

References

See also