Graphs & Trees: Intro

In Topic 4, we talked about the relation data structure, which is a set of ordered pairs of the form $(s, t)$, where $s \in S$ and $t \in T$. When $S = T$, we called it a relation on $S$.

Example: the relation $\text{GreaterThan}$ on $S = \{1, 2, 3, 4\}$ consists of the pairs $\{(2, 1),$$\; (3, 2),$$\; (3, 1),$$\; (4, 3),$$\; (4, 2),$$\; (4, 1)\}$.

Besides listing the elements of a relation, drawing a mapping diagram, or plotting the relation on the coordinate plane, we have two more ways to visualize, and hence learn more, about a relation: (1) an adjacency matrix and (2) a directed graph.

The adjacency matrix for the relation $\text{GreaterThan}$ above, for example, will be: $M = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \end{bmatrix}$.