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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:

${\displaystyle \forall a,b\in X(aRb\Leftrightarrow bRa).}$

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.

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

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.

Properties

• A symmetric and transitive relation is always quasireflexive.

• A symmetric, transitive, and reflexive relation is called an equivalence relation.

• One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge's two vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatorially equivalent objects.

See also

• Commutative property
• Symmetry in mathematics
• Symmetry