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.

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.

Results
Every relation has exactly one of these properties: it is either reflexive, antireflexive or non-reflexive.

Proof:

A reflexive relation can not be antireflexive, and vice versa:

$$\left({x, x}\right) \in \mathcal{R} \iff \lnot \left({\left({x, x}\right) \notin \mathcal{R}}\right)$$

By the definition of non-reflexive, a reflexive relation can also not be non-reflexive.

So a reflexive relation is neither antireflexive nor non-reflexive.

An antireflexive relation can be neither reflexive (see above) nor non-reflexive (by the definition of non-reflexive).

By its own definition, if a relation is neither reflexive nor antireflexive, then it is non-reflexive.

The null relation is antireflexive:

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