symmetric relation
please note:
- the text and code below is from The Pseudopedia
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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:
∀ a b ∈ X a R b ⇒ ; 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 ('preys on' in biological sciences), 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.
Graph-theoretic interpretation
In an
undirected graph, the relation over the set of
vertices of the graph under which
v and
w are related if and only if they are adjacent forms a symmetric relation. Conversely, if
R is a symmetric relation over a set
X, one can interpret it as describing an undirected graph with the elements of
X as the vertices and the pairs in
R as the edges. Thus, symmetric relations and undirected graphs are combinatorially equivalent objects.
Examples
- "is married to" is a symmetric relation, while "is less than" is not.
- "is equal to" (equality)
- "is comparable to" for a partial order
- "... is odd and ... is odd too":
missing image!
- Bothodd.png -
See also
Relació simètricaSymetrická relaceSymmetrie (Mengenlehre)Relación simétricaRelazione simmetricaיחס סימטריSzimmetrikus reláció対称関係Relacja symetrycznaСимметричное отношениеSymetrická reláciaSymmetrisk relationСиметричне відношення对称关系
- content above as imported from The Pseudopedia
- "symmetric relation" does not exist on GetWiki
- time: 10:27am EDT - Thu, Mar 18 2010
