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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:
- Failed to parse (Missing texvc executable; please see math/README to configure.): \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":
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See also
de:Symmetrie (Mengenlehre) es:Relación simétrica it:Relazione simmetrica hu:Szimmetrikus reláció ja:対称関係 pl:Relacja symetryczna ru:Симметричность sk:Symetrická relácia sv:Symmetrisk relation uk:Симетричне відношення
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