Symmetric relation

From Exampleproblems

In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.

In mathematical notation, this is:

$\forall a, b \in X,\ a R b \Rightarrow \; b R a$

For example, "is married to" is a symmetric relation, while "is less than" is not.

Note that symmetry is not the opposite of antisymmetry (aRb and bRa implies b = a). There are relations which are both symmetric and antisymmetric (equality), there are relations which are neither symmetric nor antisymmetric (divisibility), there are relations which are symmetric and not antisymmetric (congruence modulo n), and there are relations which are not symmetric but are anti-symmetric ("is less than or equal to").

A symmetric relation that is also transitive and reflexive is an equivalence relation.