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.

Example
An example of a non-reflexive relation:

Let $S = \set {a, b}, \mathcal R = \set {\paren {a, a} }$.

Then:
 * $\mathcal R$ is not reflexive, because $\paren {b, b} \notin \mathcal R$.
 * $\mathcal R$ is not antireflexive, because $\paren {a, a} \in \mathcal R$.

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

Also known as
Some sources use the term irreflexive.

However, as irreflexive is also found in other sources to mean antireflexive, it is better to use the clumsier, but less ambiguous, non-reflexive.

Also see

 * Definition:Reflexivity


 * Definition:Reflexive Relation
 * Definition:Coreflexive Relation
 * Definition:Antireflexive Relation