# Endorelation/Examples/Properties of Arbitrary Relation 3

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## 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:

## 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.

$\blacksquare$

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $9$