Definition:Non-Reflexive Relation

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

 * Reflexivity


 * Reflexive Relation
 * Coreflexive Relation
 * Antireflexive Relation