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 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