Bridge to Abstract Mathematics: Mathematical Proof and Structures

260 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8

highly non-one to one and so "do not have inverses" (i.e., their inverse

relations are not functions). The functions we know as sin-', tan-', and

so on, then, are not actually the inverses of sin and tan, respectively, but

rather are, inverses of restrictions of those functions to certain convenient

subsets of their domains.

DEFINITION 4

Let f: A B be a mapping and let X c A. We define a new mapping, the

restriction of f to X, denoted f/,, by f/,: X -* B where f/,(x) = f(x) for all x E X.

Note that, given f: A -+ B and a subset X of A, a mapping g: X -+ B

equals f/, if and only if g(x) = f (x) for all x E X. As one example, the func-

tion g(x) = fi is not the inverse of f(x) = x2, but rather, of the restriction

f/,, where X = [0, oo). Similarly, tan-' is the inverse of tan/,-,,,,,,, while

cosh- ' is the inverse of cash/,,,,,.

An important special case of restriction is the inclusion mapping. If A is

any set and X is a subset of A, we define the mapping i, by i, = (I,)/,;

that is, i,: X -, A with i,(x) = x for all x E X.

In addition to its usefulness in the definitions of certain inverse functions,

the notion of "restriction" provides mathematicians with a notational

convenience that proves valuable in a variety of settings in higher-level

mathematics..

COMPOSITION
Unlike inverse and restriction, composition of functions and mappings is
a binary operation. You have encountered this operation in both precal-
culus and elementary calculus classes and will probably recall, for instance,
warnings from instructors to understand the difference between functions
such as f(x) = (sin x)ex, a product of functions, and g(x) = sin (ex), a com-
position of the sine function with the exponential function. You will also
recall from elementary calculus that the chain rule is a rule for calculating
/
the derivative of a composition of two functions. We now give a definition
of composition from the "ordered pair" point of view.

DEFINITION 5

Let f and g be functions. Consider the relation h = {(x, z)lthere exists y such

that (x, y) E f and (y, z) ~g}. The relation h is called the composite of f and g, or

simply g composition f, and is denoted h = g 0 f.

EXAhW LE (^3) Let f = ((1,3), (2,7), (3, 101, (4, 171, (5,20)3, g = ((2,6), (3,4),
(7, lo), (17, lo), (20, lo)), and k = ((4, 3), (6, 5), (10,20)). Calculate f 0 k,
g 0 k, and k 0 g.
Solution Note first that f 0 k may have at most 15 ordered pairs, since it is
clearly a subset of (dom k) x (rng f ). But, due to the fact that f and k are