# Definition:Non-reflexive Relation

(Redirected from Definition:Non-Reflexive Relation)

## Definition

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

$\mathcal R$ is **non-reflexive** if and only if 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

- Results about
**reflexivity of relations**can be found here.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.5$: Properties of Relations