# Reflexive relation

<https://en.wikipedia.org/wiki/Reflexive_relation>

## Reflexive relation

In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself.

Formally, this may be written ∀x ∈ X . (x, x) ∈ R, or as I ⊆ R where I is the identity relation on X.

> A binary relation R on set S is **reflexive** if every element of S is related to itself, xRx or R(x,x).

R on S is reflexive if `∀a. a ∈ S -> (a,a) ∈ R`

A relation $$R$$ on the set $$\mathbb{N^+}$$ is reflexive if it contains **all diagonal (or identity) pairs** e.g. $$R={(1,1),(2,2),(3,3),\dots}$$.

```
↻◽-------◽↺
 |       |
 |       |
↻◽-------◽↺
```

A graph that represents a reflexive relation is the one where each vertex has an identity arrow.

## Irreflexive relation

> A binary relation R on set S is **irreflexive** (anti-reflexive) if no element of S is related to itself, ¬R(x,x), x is not R-related to x.

Rel R on a set S is irreflexive if `∀a. a ∈ S -> (a,a) ∉ R`

A relation $$R$$ on the set $$\mathbb{N^+}$$ is irreflexive if it contains no diagonal (or identity) pairs.

## References

<https://en.wikipedia.org/wiki/Reflexive_relation> <https://proofwiki.org/wiki/Definition:Coreflexive_Relation> <https://proofwiki.org/wiki/Definition:Relation>


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