Irreflexive relation

A relation $R$ on a set $A$ is called irreflexive if $(a,a) \notin R$ for every element $a \in A$, in effect there are no loops.

For example:

• Irreflexive: $<$, $>$, $\ne$ on $\mathbb{N}$ or on $\mathbb{Z}$ Elements do not refer to themselves.

• Irreflexive: $R_3=\{(1,2),(1,3),(2,1),(3,2)\}$ on $\{1,2,3\}$

  digraph G {
  	1->2
    1->3
    2->1
    3->2
  }

title: Not all "not reflexive" relations are irreflexive!
**Not irreflexive**: $R_2 = \{ (1,1), (1,2), (3,1) \}$ on $\{ 1,2,3 \}$
Some elements refer to themselves.
 
```graphviz
digraph G {
  1->1
  1->2
  3->1
}