Endorelation/Examples/Properties of Arbitrary Relation 1
Jump to navigation
Jump to search
Examples of Endorelation
Let $V = \set {u, v, w, x}$.
Let $E$ be the relation on $V$ defined as:
- $E = \set {\tuple {u, v}, \tuple {v, u}, \tuple {v, w}, \tuple {w, v} }$
Then $E$ is:
Proof
For all $a \in V$, we have that:
- $\tuple {a, a} \notin E$
and so $E$ is antireflexive.
For all $\tuple {a, b} \in E$, we have that:
- $\tuple {b, a} \in E$
and so $E$ is symmetric.
We have:
- $\tuple {u, v} \in E$ and $\tuple {v, w} \in E$, but not $ \tuple {u, w} \in E$
and so $E$ is not non-transitive.
$\blacksquare$
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Example $\text{A}.1$