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# reflexive relation

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.

Defines:

reflexivity, reflexive

Related:

Symmetric, Transitive3, Antisymmetric, Irreflexive

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E20*no label found*

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