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Symmetric relation

Type of binary relation

Symmetric relation
Summary

Type of binary relation

A symmetric relation is a type of binary relation. Formally, a binary relation R over a set X is symmetric if: : \forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation aRb means that (a, b) ∈ R.

An example is the relation "is equal to", because if is true then is also true. If RT represents the converse of R, then R is symmetric if and only if .

Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.

Examples

In mathematics

  • "is equal to" (equality) (whereas "is less than" is not symmetric)
  • "is comparable to", for elements of a partially ordered set
  • "... and ... are odd": ::::::[[Image:Bothodd.png]]

Outside mathematics

  • "is married to" (in most legal systems)
  • "is a fully biological sibling of"
  • "is a homophone of"
  • "is a co-worker of"
  • "is a teammate of"

Relationship to asymmetric and antisymmetric relations

Symmetric and antisymmetric relations

By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").

Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if ) are actually independent of each other, as these examples show.

Not antisymmetriccongruence in modular arithmetic// (integer division), most nontrivial permutations
Not antisymmetricis a full biological sibling ofpreys on

Properties

  • A symmetric and transitive relation is always quasireflexive.
  • One way to count the symmetric relations on n elements, that in their binary matrix representation the upper right triangle determines the relation fully, and it can be arbitrary given, thus there are as many symmetric relations as n × n binary upper triangle matrices, 2n(n+1)/2.

Notes

References

References

  1. "MAD3105 1.2". Florida State University.
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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