Endorelation/Examples/Properties of Arbitrary Relation 2
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Examples of Endorelation
Let $V = \set {a, b, c, d}$.
Let $R$ be the relation on $V$ defined as:
- $E = \set {\tuple {a, a}, \tuple {a, b}, \tuple {a, c}, \tuple {a, d}, \tuple {b, b}, \tuple {b, c}, \tuple {b, d}, \tuple {c, c}, \tuple {c, d}, \tuple {d, d} }$
Then $E$ is:
Proof
For all $x \in V$, we have that:
- $\tuple {x, x} \in R$
and so $R$ is reflexive.
For all $\tuple {x, y} \in R$, we have that:
- $\tuple {y, x} \in R \iff x = y$
and so $R$ is antisymmetric.
We have:
- $\tuple {x, y} \in R$ and $\tuple {y, z} \in R$ implies that $ \tuple {x, z} \in R$
and so $R$ is not transitive.
$\blacksquare$
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $8$