Symmetric relation

A symmetric relation is a type of binary relation. A homogeneous relation $R$ on a set $X$ is symmetric if:Template:Refn

for all $a$ and $b$ in $X$, $aRb$ if and only if $bRa$,

where the notation $aRb$ means that the ordered pair $(a,b)$ is in the relation $R$.

An example is the relation "is equal to", because if Template:Nowrap is true then Template:Nowrap is also true. If R^T represents the converse of R, then R is symmetric if and only if Template:Nowrap.

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.Template:Refn

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"
• "has coauthored a book with"
• "is a homophone of"
• "is a co-worker of"
• "is a 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 Template:Nowrap) 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.Template:Efn
• 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 Template:Nowrap binary upper triangle matrices, 2^n(n+1)/2.Template:Refn

See also

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