Bridge to Abstract Mathematics: Mathematical Proof and Structures

262 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8

In view of Theorem 2, we will dispense henceforth with ordered pair

notation in discussing the composition of two functions, using functional

notation instead.

The final property of function composition suggested by the calculations

from Example 3 is associativity, with the following precise statement:

THEOREM 3

If f, g, and h are functions, then f 0 (g o h) = (f 0 g) o h.

Proof By Exercise 2, we may prove two functions equal by showing that

(1) their domains are equal and (2) their values are the same at each point

in the common domain.

1. Let x be an element of dorn [( f 0 g) 0 h)]. Then x E dorn h and
   h(x) E dorn (f 0 g). The latter property means that h(x) E dorn g and
   g(h(x)) E dorn f. But since x E dorn h and h(x) E dorn g, then x E
   dorn (g 0 h). Since x E dorn (g 0 h) and (g 0 h)(x) E dorn f, then
   x E dorn [ f 0 (g 0 h)], as desired. Hence dorn [( f 0 g) 0 h] c
   dorn [ f 0 (g 0 h)]. The proof of the reverse inclusion is completely
   analogous.


2. Let x E dorn [( f 0 g) 0 h] ( = dorn [ f 0 (g 0 h)]). Then ( f 0 g) 0 h =
   ((f O g)(h(x))) = f (g(h(x))) = f ((g O h)(x)) = (f 0 g) 0 h. since
   (f 0 g) 0 h = f 0 (g 0 h) for any value of x in the common
   domain of the two functions f 0 (g 0 h) and (f 0 g) 0 h, we conclude
   that these two functions are actually the same, as desired. 0

Suppose now that f: A + B and g: B -, C are mappings, where we make
explicit note of the assumption that the codomain of the first mapping
equals the domain of the second. Due to this assumption, we have that
for any x E A = dorn f, f (x) E B = dorn g. From this, we may easily con-
clude that the function g 0 f has the domain A and a range that is a subset
of C. Therefore it is meaningful to refer to the mapping g 0 f: A + C when-
ever we are given mappings f: A + B and g: B --+ C. By common agree-
/. ment among mathematicians, this is the only circumstance in which we
/“ consider the composition of two mappings [see, however, Exercise Iqe)].

This situation is often pictured by diagrams such as the one in Figure 8.4.

There are other fairly standard ways of creating new mappings from old

ones, in addition to inverse, restriction, and composition. Several examples

are provided in Exercises 11, 12, and 13.

Figure 8.4 A diagrammatic view of the

composition of mappings.