Definition:Reflexivity

Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

Reflexive
$\mathcal R$ is reflexive iff:


 * $\forall x \in S: \left({x, x}\right) \in \mathcal R$

Coreflexive
$\mathcal R$ is coreflexive (pronounced co-reflexive, not core-flexive) iff:


 * $\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$

Antireflexive
$\mathcal R$ is antireflexive (or irreflexive) iff:


 * $\forall x \in S: \left({x, x}\right) \notin \mathcal R$

Non-reflexive
$\mathcal R$ is non-reflexive iff it is neither reflexive nor antireflexive.

An example of a non-reflexive relation:

Let $S = \left\{{a, b}\right\}, \mathcal R = \left\{{\left({a, a}\right)}\right\}$.


 * $\mathcal R$ is not reflexive, because $\left({b, b}\right) \notin \mathcal R$.
 * $\mathcal R$ is not antireflexive, because $\left({a, a}\right) \in \mathcal R$.

So being neither one thing nor the other, it must be non-reflexive.

Also see

 * Symmetry
 * Transitivity