Symmetric relation - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of

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). See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if 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"). Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without See Congruence (geometry for the term as used in elementary geometry In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

Properties containing the symmetric relation

equivalence relation - A symmetric relation that is also transitive and reflexive. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b 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)
". Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric . . is odd and . . . is odd too":