# 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.

 Title reflexive relation Canonical name ReflexiveRelation Date of creation 2013-03-22 12:15:36 Last modified on 2013-03-22 12:15:36 Owner yark (2760) Last modified by yark (2760) Numerical id 17 Author yark (2760) Entry type Definition Classification msc 03E20 Related topic Symmetric Related topic Transitive3 Related topic Antisymmetric Related topic Irreflexive Defines reflexivity Defines reflexive