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.