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