PlanetMath: symmetric relation

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symmetric relation

(Definition)

A relation $\mathcal{R}$ on a set is symmetric if and only if whenever $x\mathcal{R}y$ for some $x, y \in A$ then also $y\mathcal{R}x$ .

An example of a symmetric relation on $\{a,b,c\}$ is $\{(a,a), (c,b), (b,c), (a,c), (c,a)\}$ . One relation that is not symmetric is $\mathcal{R} = \{(b,b), (a,b), (b,a), (c,b) \}$ , because $(c,b) \in \mathcal{R}$ but $(b,c) \notin \mathcal{R}$ .

On a finite set with elements there are $2^{n^2}$ relations, of which $2^{\frac{n^2+n}{2}}$ are symmetric.

A relation $\mathcal{R}$ that is both symmetric and antisymmetric has the property that $x\mathcal{R}y$ implies . On a finite set with elements there are only such relations.

"symmetric relation" is owned by yark. [ full author list (3) | owner history (2) ]

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Also defines:

symmetry, symmetric

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Cross-references: antisymmetric, finite set, relation
There are 38 references to this entry.

This is version 17 of symmetric relation, born on 2002-02-02, modified 2006-10-19.
Object id is 1647, canonical name is Symmetric.
Accessed 13072 times total.

Classification:

AMS MSC:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

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