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Applied Discrete Structures

Al Doerr, Ken Levasseur

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Section 6.2 Graphs of Relations on a Set

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In this section we introduce directed graphs as a way to visualize relations on a set.
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Subsection 6.2.1 Digraphs

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Let A={0,1,2,3}, and let
r={(0,0),(0,3),(1,2),(2,1),(3,2),(2,0)}
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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 arb. This type of graph of a relation r is called a directed graph or digraph. Figure 6.2.1 is a digraph for r. Notice that since 0 is related to itself, we draw a β€œself-loop” at 0.
Digraph of the relation \(r = \{(0, 0), (0, 3), (1, 2), (2, 1), (3, 2), (2, 0)\}\) with four nodes, 0, 1, 2, and 3; and edges between them corresponding to the pairs in the relation. One pair, \((0,0)\) is displayed as a loop starting at ending at node 0.
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Figure 6.2.1. Digraph of a relation
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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.
Digraph of the relation r, alternate embedding
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Figure 6.2.2. Alternate embedding of the previous directed graph
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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.
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)}.
Digraph of the relation \(s\)
Figure 6.2.4. Digraph of the relation s
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We will be building on the next example in the following section.
Let B={1,2}, and let A=P(B)={βˆ…,{1},{2},{1,2}}. Then βŠ† is a relation on A whose digraph is Figure 6.2.6.
Graph for set containment on subsets of \(\{1,2\}\)
Figure 6.2.6. Graph for set containment on subsets of {1,2}
We will see in the next section that since βŠ† has certain structural properties that describe β€œpartial orderings.” We will be able to draw a much simpler type graph than this one, but for now the graph above serves our purposes.
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Exercises 6.2.2 Exercises

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1.

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Let A={1,2,3,4}, and let r be the relation ≀ on A. Draw a digraph for r.
Answer.
Digraph for \(\leq\)
Figure 6.2.7.
Hasse diagram for \(\leq\)
Figure 6.2.8.
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2.

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Let B={1,2,3,4,6,8,12,24}, and let s be the relation β€œdivides” on B. Draw a digraph for s.
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3.

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Let A={1,2,3,4,5}. Define t on A by atb if and only if bβˆ’a is even. Draw a digraph for t.
Answer.
Figure 6.2.9. Digraph of the relation t
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4.

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Let A be the set of strings of 0’s and 1’s of length 3 or less. This includes the empty string, Ξ», which is the only string of length zero.
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    Define the relation of d on A by xdy if x is contained within y. For example, 01d101. Draw a digraph for this relation.
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    Do the same for the relation p defined by xpy if x is a prefix of y. For example, 10p101, but 01p101 is false.
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5.

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Recall the relation in Exercise 5 of Section 6.1, ρ defined on the power set, P(S), of a set S. The definition was (A,B)∈ρ iff A∩B=βˆ…. Draw the digraph for ρ where S={a,b}.
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6.

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Let C={1,2,3,4,6,8,12,24} and define t on C by atb if and only if a and b share a common divisor greater than 1. Draw a digraph for t.
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