Relation/Examples/Ordering on Arbitrary Sets of Integers

Example of Relation
Let $A = \set {1, 2, 3, 4}$ and $B = \set {1, 2, 3}$ be sets of integers.

Consider the following diagram, where:


 * $A$ runs along the top
 * $B$ runs down the left hand side
 * a relation $\mathcal R$ between $A$ and $B$ is indicated by marking with $\bullet$ every ordered pair $\tuple {a, b} \in A \times B$ which is in the truth set of $\mathcal R$


 * $\begin{array}{r|rrrr}

A \times B & 1 & 2 & 3 & 4 \\ \hline 1 & \bullet & \bullet & \bullet & \circ \\ 2 & \bullet & \bullet &  \circ & \circ \\ 3 & \bullet &  \circ &   \circ & \circ \\ \end{array}$

This relation $\mathcal R$ can be described as:
 * $\mathcal R = \set {\tuple {x, y} \in A \times B: x + y \le 4}$