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Symmetric relation

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Title: Symmetric relation  
Author: World Heritage Encyclopedia
Language: English
Subject: Preorder, Binary relation, Asymmetric relation, Equivalence relation, Antisymmetric relation
Collection: Mathematical Relations, Symmetry
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Symmetric relation

In mathematics and other areas, 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.


Contents

  • Examples 1
    • In mathematics 1.1
    • Outside mathematics 1.2
  • Relationship to asymmetric and antisymmetric relations 2
  • Additional aspects 3
  • See also 4

Examples

In mathematics

Outside mathematics

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

Relationship to asymmetric and antisymmetric relations

By definition, a 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 a = b) are actually independent of each other, as these examples show.

Mathematical examples
Symmetric Not symmetric
Antisymmetric equality "is less than or equal to"
Not antisymmetric congruence in modular arithmetic "is divisible by", over the set of integers
Non-mathematical examples
Symmetric Not symmetric
Antisymmetric "is the same person as, and is married" "is the plural of"
Not antisymmetric "is a full biological sibling of" "preys on"

Additional aspects

A symmetric relation that is also transitive and reflexive is an equivalence relation.

One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge's two vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatorially equivalent objects.

See also

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