# 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 mathematical notation, this is:

:$forall a, b in X, a R b Rightarrow ; b R a.$

Note: symmetry is not the exact opposite of "antisymmetry" ("aRb" and "bRa" implies "b" = "a"). There are relations which are both symmetric and antisymmetric (equality and its subrelations, including, vacuously, the empty relation), 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 antisymmetric ("is less than or equal to").

Properties containing the symmetric relation

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

Examples

* "is married to" is a symmetric relation, while "is less than" is not.
* "is equal to" (equality)
* "... is odd and ... is odd too":

::::::

ee also

* Symmetry in mathematics.
* nonsymmetric relation
* asymmetric relation
* antisymmetric relation

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