6.2: Graphs of Relations on a Set

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

Digraphs

Let $A = \{0, 1,2,3\}\text{,}$ and let

$$\begin{equation*} r = \{(0, 0), (0, 3), (1, 2), (2, 1), (3, 2), (2, 0)\} \end{equation*}$$

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 $a r b\text{.}$ This type of graph of a relation $r$ is called a directed graph or digraph. Figure $\PageIndex{1}$ is a digraph for $r\text{.}$ Notice that since 0 is related to itself, we draw a “self-loop” at 0.

Figure $\PageIndex{1}$: Digraph of a relation

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 $\PageIndex{1}$ could also be presented as in Figure $\PageIndex{2}$.

Figure $\PageIndex{2}$: Alternate embedding of the previous directed graph

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 the two graphs.

Example $\PageIndex{1}$: Another Directed Graph

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

Figure $\PageIndex{3}$: Digraph of the relation $s$

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

Example $\PageIndex{2}$: Ordering Subsets of a Two Element Universe

Let $B = \{1,2\}\text{,}$ and let $A = \mathcal{P}(B) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}\text{.}$ Then $\subseteq$ is a relation on $A$ whose digraph is Figure $\PageIndex{4}$.

Figure $\PageIndex{4}$: Graph for set containment on subsets of $\{1,2\}$

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.

 Exercises

Exercise $\PageIndex{1}$

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

Answer

Figure $\PageIndex{5}$: Digraph for $\leq$

Figure $\PageIndex{6}$: Hasse diagram for $\leq$

Exercise $\PageIndex{2}$

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

Exercise $\PageIndex{3}$

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

Answer

See Figure $\PageIndex{7}$

Figure $\PageIndex{7}$: Digraph of the relation $t$

Exercise $\PageIndex{4}$

Let $A$ be the set of strings of 0's and 1's of length 3 or less. This includes the empty string, $\lambda\text{,}$ which is the only string of length zero.

1. Define the relation of $d$ on $A$ by $x d y$ if $x$ is contained within $y\text{.}$ For example, $01 d 101\text{.}$ 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, $10 p 101\text{,}$ but $01 p 101$ is false.

Exercise $\PageIndex{5}$

Recall the relation in Exercise 6.1.5 of Section 6.1, $\rho$ defined on the power set, $\mathcal{P}(S)\text{,}$ of a set $S\text{.}$ The definition was $(A,B) \in \rho$ iff $A\cap B = \emptyset\text{.}$ Draw the digraph for $\rho$ where $S = \{a, b\}\text{.}$

Exercise $\PageIndex{6}$

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 $t\text{.}$