# Endorelation/Examples/Properties of Arbitrary Relation 3

## Examples of Endorelation

Let $V = \set {a, b, c, d}$.

Let $R$ be the relation on $V$ defined as:

$r = \set {\tuple {a, a}, \tuple {a, b}, \tuple {b, b}, \tuple {b, c}, \tuple {c, b}, \tuple {c, c} }$

Then $E$ is:

non-reflexive
non-symmetric
non-transitive.

## Proof

We have that:

$\tuple {d, d} \notin R$

and so $R$ is not reflexive.

We also have that:

$\tuple {a, b} \in R$

but:

$\tuple {b, a} \notin R$

and so $R$ is not symmetric.

We have that:

$\tuple {b, c} \in R$

but:

$\tuple {c, b} \notin R$

and so $R$ is not asymmetric.

It follows that $R$ is non-symmetric.

We have:

$\tuple {a, b} \in R$ and $\tuple {b, c} \in R$

but we also have that:

$\tuple {a, c} \notin R$

and so $R$ is not non-transitive.

$\blacksquare$