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## Section6.2Graphs of Relations on a Set

In this section we introduce directed graphs as a way to visualize relations on a set.

### Subsection6.2.1Digraphs

Let $A = \{0, 1,2,3\}$, and let $r = \{(0, 0), (0, 3), (1, 2), (2, 1), (3, 2), (2, 0)\}$ In representing this relation as a graph, elements of $A$ are called the vertices of the graph. They are typically represented by labeled points or small circles. We connect vertex $a$ to vertex $b$ with an arrow, called an edge, going from vertex $a$ to vertex $b$ if and only if $a r b$. This type of graph of a relation $r$ is called a directed graph or digraph. Figure 6.2.1 is a digraph for $r\text{.}$ Notice that since 0 is related to itself, we draw a “self-loop” at 0.

The actual location of the vertices in a digraph is immaterial. The actual location of vertices we choose is called an embedding of a graph. The main idea is to place the vertices in such a way that the graph is easy to read. After drawing a rough-draft graph of a relation, we may decide to relocate the vertices so that the final result will be neater. Figure 6.2.1 could also be presented as in Figure 6.2.2.

A vertex of a graph is also called a node, point, or a junction. An edge of a graph is also referred to as an arc, a line, or a branch. Do not be concerned if two graphs of a given relation look different as long as the connections between vertices are the same in two graphs.

Consider the relation $s$ whose digraph is Figure 6.2.4. What information does this give us? The graph tells us that $s$ is a relation on $A = \{1, 2, 3\}$ and that $s = \{(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 3)\}$.

We will be building on the next example in the following section.

Let $B = \{1,2\}$, and let $A = \mathcal{P}(B) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$. Then $\subseteq$ is a relation on $A$ whose digraph is Figure 6.2.6.

We will see in the next section that since $\subseteq$ has certain structural properties that describe “partial orderings.” We will be able to draw a much simpler type graph than this one, but for now the graph above serves our purposes.

### SubsectionExercises

###### 1

Let $A = \{1, 2, 3, 4\}$, and let $r$ be the relation $\leq$ on $A\text{.}$ Draw a digraph for $r\text{.}$

###### 2

Let $B = \{1,2, 3, 4, 6, 8, 12, 24\}$, and let $s$ be the relation “divides” on $B\text{.}$ Draw a digraph for $s\text{.}$

###### 3

Let $A=\{1,2,3,4,5\}$. Define $t$ on $A$ by $a t b$ if and only if $b - a$ is even. Draw a digraph for $t\text{.}$

1. Let $A$ be the set of strings of 0's and 1's of length 3 or less. Define the relation of $d$ on $A$ by $x d y$ if $x$ is contained within $y\text{.}$ For example, $01 d 101$. Draw a digraph for this relation.
2. Do the same for the relation $p$ defined by $x p y$ if $x$ is a prefix of $y\text{.}$ For example, $10 p 101$, but $01 p 101$ is false.
Recall the relation in Exercise 5 of Section 6.1, $\rho$ defined on the power set, $\mathcal{P}(S)$, of a set $S\text{.}$ The definition was $(A,B) \in \rho$ iff $A\cap B = \emptyset$. Draw the digraph for $\rho$ where $S = \{a, b\}$.
Let $C= \{1,2, 3, 4, 6, 8, 12, 24\}$ and define $t$ on $C$ by $a t b$ if and only if $a$ and $b$ share a common divisor greater than 1. Draw a digraph for $t\text{.}$