# Non-reflexive Relation/Examples

## Examples of Non-reflexive Relations

### 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 $\mathcal R$ be the relation on $S$ defined as:

$\mathcal R = \set {\paren {a, a} }$

Then $\mathcal R$ is neither:

a reflexive relation, as $\tuple {b, b} \notin \mathcal R$

nor:

an antireflexive relation, as $\tuple {a, a} \in \mathcal R$

Thus $\mathcal R$ is a non-reflexive relation.