Page 187 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Relations 173
Finally, suppose R is a relation from A to B and S is a relation from B to C.
Then the composition of S and R is the relation S ◦ R from A to C defined as
follows:
S ◦ R ={(a, c) ∈ A × C |∃b ∈ B((a, b) ∈ R and (b, c) ∈ S)}.

Notice that we have assumed that the second coordinates of pairs in R and the
first coordinates of pairs in S both come from the same set, B. If these sets were
not the same, the composition S ◦ R would be undefined.

According to Definition 4.2.3, the domain of a relation from A to B is the
set containing all the first coordinates of ordered pairs in the relation. This will
in general be a subset of A, but it need not be all of A. For example, consider
the relation L from part 4 of Example 4.2.2, which pairs up students with the
dorm rooms in which they live. The domain of L would contain all students
who appear as the first coordinate in some ordered pair in L – in other words,
all students who live in some dorm room – but would not contain, for example,
students who live in apartments off campus. Working it out more carefully from
the definition as stated, we have

Dom(L) ={s ∈ S |∃r ∈ R((s,r) ∈ L)}
={s ∈ S |∃r ∈ R (the student s lives in the dorm room r)}
={s ∈ S | the student s lives in some dorm room}.

Similarly, the range of a relation is the set containing all the second coordinates
of its ordered pairs. For example, the range of the relation L would be the set
of all dorm rooms in which some student lives. Any dorm rooms that are
unoccupied would not be in the range of L.
The inverse of a relation contains exactly the same ordered pairs as the
original relation, but with the order of the coordinates of each pair reversed.
Thus, in the case of the relation L, if Joe Smith lives in room 213 Davis Hall,
then (Joe Smith, 213 Davis Hall) ∈ L and (213 Davis Hall, Joe Smith) ∈ L −1 .
In general, for any student s and dorm room r, we would have (r, s) ∈ L −1
iff (s,r) ∈ L. For another example, consider the relation G from part 2 of
Example 4.2.2. It contains all ordered pairs of real numbers (x, y) in which x
is greater than y. We might call it the “greater-than” relation. Its inverse is
G −1 ={(x, y) ∈ R × R | (y, x) ∈ G}
={(x, y) ∈ R × R | y > x}
={(x, y) ∈ R × R | x < y}.

In other words, the inverse of the greater-than relation is the less-than relation!