Endorelation/Examples/Properties of Arbitrary Relation 3
Jump to navigation
Jump to search
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 $\text{A}.2$: $9$