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:

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

Note: symmetry is not the exact opposite of antisymmetry (aRb and bRa implies b = a). There are relations which are both symmetric and antisymmetric (for example, 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"). Also, the empty relation is, vacuously, both symmetric and asymmetric.

Properties containing the symmetric relation

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

Examples

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

  See also

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  • Symmetry in mathematics.

  Set theory | Symmetry

  Symetrická relace | Symmetrie (Mengenlehre) | Relación simétrica | Szimmetrikus reláció | Relazione simmetrica | Relacja symetryczna | Symetrická relácia | Symmetrisk relation | 对称关系