# Reflexive Relation

## 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.

### Examples

• In the set $$\mathbb {N}$$ 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 $$\mathbb {N}^{\ast}$$ of non-zero whole numbers, the relation “…divides…” is a reflexive relation.
• In the set $$\mathbb {N}^{\ast}$$ of non-zero whole numbers, the relation “…is relatively prime with…” is an anti-reflexive relation.