Endorelation/Examples/Properties of Arbitrary Relation 2

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:

reflexive
antisymmetric
transitive.

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.

$\blacksquare$