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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Relation (mathematics) — This article sets out the set theoretic notion of relation. For a more elementary point of view, see binary relations and triadic relations. : For a more combinatorial viewpoint, see theory of relations. In mathematics, especially set theory, and …   Wikipedia

  • symmetric — adjective a) Symmetrical. Is a sibling of is a symmetric relation. b) Of a relation R on a set S, such that xRy if and only if yRx for all members x and y of S (that is, if the relation holds between any element and a second, it also holds… …   Wiktionary

  • Relation algebra — is different from relational algebra, a framework developed by Edgar Codd in 1970 for relational databases. In mathematics, a relation algebra is a residuated Boolean algebra supporting an involutary unary operation called converse. The… …   Wikipedia

  • Symmetric polynomial — This article is about individual symmetric polynomials. For the ring of symmetric polynomials, see ring of symmetric functions. In mathematics, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the… …   Wikipedia

  • Symmetric group — Not to be confused with Symmetry group. A Cayley graph of the symmetric group S4 …   Wikipedia

  • Symmetric difference — Venn diagram of The symmetric difference is the union without the intersection …   Wikipedia

  • Symmetric product of an algebraic curve — In mathematics, the n fold symmetric product of an algebraic curve C is the quotient space of the n fold cartesian product: C times; C times; ... times; C or C n by the group action of the symmetric group on n letters permuting the factors. It… …   Wikipedia

  • relation — Philosophically relations are interesting because of the historic prejudice, given its most forceful expression by Leibniz, that they are somehow ‘unreal’ compared to the intrinsic, monadic properties of things. A way of putting the idea is that… …   Philosophy dictionary

  • symmetric — A relation is symmetric when if R xy, then R yx . For example, if a person x has the same parents as y, then y has the same parents as x …   Philosophy dictionary

  • symmetric — /səˈmɛtrɪk/ (say suh metrik) adjective 1. → symmetrical. 2. Logic, Mathematics denoting a relation, such that if it is valid between a and b it is valid between b and a, as: a = b implies b = a …   Australian English dictionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.