Page 195 - HOW TO PROVE IT: A Structured Approach, Second Edition

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More About Relations 181
Another way to think about relations is to draw pictures of them. Figure 1
shows a picture of the relation R ={(1, 3), (1, 5), (3, 3)} from part 1 of Exam-
ple 4.2.2. Recall that this was a relation from the set A ={1, 2, 3} to the set
B ={3, 4, 5}. In the figure, each of these sets is represented by an oval, with
the elements of the set represented by dots inside the oval. Each ordered pair
(a, b) ∈ R is represented by an arrow from the dot representing a to the dot
representing b. For example, there is an arrow from the dot inside A labeled 1
to the dot inside B labeled 5 because the ordered pair (1, 5) is an element of R.
In general, any relation R from a set A to a set B can be represented by such a
picture.ThedotsrepresentingtheelementsofAandBinsuchapicturearecalled
vertices, and the arrows representing the ordered pairs in R are called edges.It
doesn’t matter exactly how the vertices representing elements of A and B are
arranged on the page; what’s important is that the edges correspond precisely
to the ordered pairs in R. Drawing these pictures may help you understand
the concepts discussed in the last section. For example, you should be able
to convince yourself that you could find the domain of R by locating those
vertices in A that have edges pointing away from them. Similarly, the range
of R would consist of those elements of B whose vertices have edges pointing
toward them. For the relation R shown in Figure 1, we have Dom(R) ={1, 3}
and Ran(R) ={3, 5}. A picture of R −1 would look just like a picture of R but
with the directions of all the arrows reversed.

Figure 1

Pictures illustrating the composition of two relations are a little harder to
understand. For example, consider again the relations E and T from parts 5
and 6 of Example 4.2.2. Figure 2 shows what part of both relations might look
like. (The complete picture might be quite large if there are many students,