Page 188 - HOW TO PROVE IT: A Structured Approach, Second Edition

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174 Relations
The most difficult concept introduced in Definition 4.2.3 is the concept of
the composition of two relations. For an example of this concept, consider
the relations E and T from parts 5 and 6 of Example 4.2.2. Recall that E
is a relation from the set S of all students to the set C of all courses, and
T is a relation from C to the set P of all professors. According to Defini-
tion 4.2.3, the composition T ◦ E will be the relation from S to P defined as
follows:

T ◦ E ={(s, p) ∈ S × P |∃c ∈ C((s, c) ∈ E and (c, p) ∈ T )}
={(s, p) ∈ S × P |∃c ∈ C(the student s is enrolled in the course c

and the course c is taught by the professor p)}
={(s, p) ∈ S × P | the student s is enrolled in some course
taught by the professor p}.

Thus, if Joe Smith is enrolled in Biology 12 and Biology 12 is taught by
Professor Evans, then (Joe Smith, Biology 12) ∈ E and (Biology 12, Professor
Evans) ∈ T , and therefore (Joe Smith, Professor Evans) ∈ T ◦ E. In general, if
s is some particular student and p is a particular professor, then (s, p) ∈ T ◦ E
iff there is some course c such that (s, c) ∈ E and (c, p) ∈ T . This notation may
seem backward at first. If (s, c) ∈ E and (c, p) ∈ T , then you might be tempted
to write (s, p) ∈ E ◦ T , but according to our definition, the proper notation is
(s, p) ∈ T ◦ E. In fact, E ◦ T is undefined, because the second coordinates of
ordered pairs in T and the first coordinates of pairs in E do not come from
the same set. The reason we’ve chosen to write compositions of relations in
this way will become clear in Chapter 5. For the moment, you’ll just have
to be careful about this notational detail when working with compositions of
relations.

Example 4.2.4. Let S, R, C, and P be the sets of students, dorm rooms,
courses, and professors at your school, as before, and let L, E, and T be
the relations defined in parts 4–6 of Example 4.2.2. Describe the following
relations.
1. E −1 .
2. E ◦ L −1 .
3. E −1 ◦ E.
4. E ◦ E −1 .
5. T ◦ (E ◦ L −1 ).
6. (T ◦ E) ◦ L −1 .