Reflexive Relation

Relation defined in a set E so that all elements x of E are related to themselves.

The arrow diagram of a reflexive relation in a set E includes loops in each of its points.
A relation in a set E that does not contain any loops is called anti-reflexive while a relation in E that is neither reflexive nor anti-reflexive is called non-reflexive.

In the set [latex]\mathbb {N}[/latex] of whole numbers, the relation "...divides..." is a non-reflexive relation. In its arrow diagram, we find loops in all of its points except 0, because 0 cannot be divided by itself.
In the set [latex]\mathbb {N}^{\ast}[/latex] of non-zero whole numbers, the relation "...divides…" is a reflexive relation.
In the set [latex]\mathbb {N}^{\ast}[/latex] of non-zero whole numbers, the relation "...is relatively prime with…" is an anti-reflexive relation.

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