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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

In mathematical notation, this is: Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

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"). In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... Vacuous truth is a special topic of first-order logic. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

Properties containing the symmetric relation

equivalence relation - A symmetric relation that is also transitive and reflexive. In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ... In set theory, a binary relation can have, among other properties, reflexivity or irreflexivity. ...

Examples

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

In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... Image File history File links No higher resolution available. ...

See also

• Symmetry in mathematics.