Rank Function/Examples/Arbitrary Example 1

Example of Rank Function
Let $X = \set {x, y, z}$.

Let $\RR$ be the relation on $X$ defined as:
 * $\RR = \set {\tuple {x, x}, \tuple {x, y}, \tuple {x, z}, \tuple {y, y}, \tuple {z, z} }$.

Let the mapping $\operatorname {rk}_0: X \to \N$ be defined as:

Then $\operatorname {rk}_0$ is a rank function for $\RR$.

Proof
There are two ordered pairs $\tuple {a, b}$ in $\RR$ such that $a \ne b$:

and the fact that $\operatorname {rk}_0$ is a rank function for $\RR$ is clear.