Page 242 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: Oyk/
0521861241c05 CB996/Velleman October 19, 2005 0:16 0 521 86124 1 Char Count= 0

228 Functions
For example, for the function F ={(1, 5), (2, 4), (3, 5)} in part 1 of the last
example, we could say that F(1) = 5, since (1, 5) ∈ F. Similarly, F(2) = 4
and F(3) = 5. If L is the function in part 3 and c is any city, then L(c) would be
the unique country n such that (c, n) ∈ L. In other words, L(c) = the country
in which c is located. For example, L(Paris) = France. For the function D in
part 5, we could say that for any person p, D(p) = the set of all children of p.
If A is any set and a ∈ A, then (a, a) ∈ i A ,so i A (a) = a. And if f is the function
2
in part 7, then for every real number x, f (x) = x .
A function f from a set A to another set B is often specified by giving a rule
that can be used to determine f (a) for any a ∈ A. For example, if A is the set
of all people and B = R , then we could define a function f from A to B by the
+
rule that for every a ∈ A, f (a) = a’s height in inches. Although this definition
doesn’t say explicitly which ordered pairs are elements of f, we can determine
this by using our rule that for all a ∈ A and b ∈ B, (a, b) ∈ f iff b = f (a).
Thus,

f ={(a, b) ∈ A × B | b = f (a)}
={(a, b) ∈ A × B | b = a’s height in inches}.

For example, if Joe Smith is 68 inches tall, then (Joe Smith, 68) ∈ f and
f(Joe Smith) = 68.
It is often useful to think of a function f from A to B as representing a rule
that associates, with each a ∈ A, some corresponding object b = f (a) ∈ B.
However, it is important to remember that although a function can be defined
by giving such a rule, it need not be defined in this way. Any subset of A × B
that satisfies the requirements given in Definition 5.1.1 is a function from
A to B.

Example 5.1.3. Here are some more examples of functions defined by rules.

1. Suppose every student is assigned an academic advisor who is a professor.
Let S be the set of students and P the set of professors. Then we can define
a function f from S to P by the rule that for every student s, f (s) = the
advisor of s. In other words,

f ={(s, p) ∈ S × P | p = f (s)}
={(s, p) ∈ S × P | the professor p is the academic advisor
of the student s}.