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Home > Symmetric relation

 
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 notation, this is:

∀ a, b ∈ X, aRb ⇒ bRa.

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" ).

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

Set theory

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