# Definition:Non-reflexive Relation

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## 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.

## 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**.

## Examples

### Arbitrary Non-reflexive Relation 1

Let $V_1 = \set {y, z}$.

Let $S$ be the relation on $V_1$ defined as:

- $S = \set {\tuple {y, y}, \tuple {y, z} }$

Then $S$ is neither:

- a reflexive relation, as $\tuple {z, z} \notin S$

nor:

- an antireflexive relation, as $\tuple {y, y} \in S$

Thus $S$ is a non-reflexive relation.

### Arbitrary Non-reflexive Relation 2

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

Let $\RR$ be the relation on $S$ defined as:

- $\RR = \set {\paren {a, a} }$

Then $\RR$ is neither:

- a reflexive relation, as $\tuple {b, b} \notin \RR$

nor:

- an antireflexive relation, as $\tuple {a, a} \in \RR$

Thus $\RR$ is a non-reflexive relation.

## 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 - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations