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reflexive relation (Definition)

A relation $ \mathcal{R}$ on a set $ A$ is reflexive if and only if $ a\mathcal{R}a$ for all $ a\in A$.

For example, let $ A = \{1,2,3\}$. Then $ \{(1,1), (2,2), (3,3), (1,3), (3,2)\}$ is a reflexive relation on $ A$, because it contains $ (a,a)$ for all $ a \in A$. However, $ \{(1,1), (2,2), (2,3), (3,1)\}$ is not reflexive because it does not contain $ (3,3)$.

On a finite set with $ n$ elements there are $ 2^{n^2}$ relations, of which $ 2^{n^2-n}$ are reflexive.



"reflexive relation" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: symmetric relation, transitive relation, antisymmetric, irreflexive

Also defines:  reflexivity, reflexive
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Cross-references: finite set, relation
There are 24 references to this entry.

This is version 13 of reflexive relation, born on 2002-02-02, modified 2006-10-19.
Object id is 1644, canonical name is Reflexive.
Accessed 10582 times total.

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