Definition:Reflexivity

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

Reflexive
$$\mathcal{R}$$ is reflexive iff:


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

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.

Example
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.