Page 246 - HOW TO PROVE IT: A Structured Approach, Second Edition

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232 Functions
derived in Theorem 5.1.5. Note that because functions are just relations of a
special kind, everything we have proven about composition of relations ap-
plies to composition of functions. In particular, by Theorem 4.2.5, we know
that composition of functions is associative.

Example 5.1.6. Here are some examples of compositions of functions.
1. Let C and N be the sets of all cities and countries, respectively, and let
L : C → N be the function defined in part 3 of Example 5.1.2. Thus, for
every city c, L(c) = the country in which c is located. Let B be the set
of all buildings located in cities, and define F : B → C by the formula
F(b) = the city in which the building b is located. Then L ◦ F : B → N.
For example, F(Eiffel Tower) = Paris, so according to the formula derived
in Theorem 5.1.5,
(L ◦ F)(Eiffel Tower) = L(F(Eiffel Tower))

= L(Paris) = France.
In general, for every building b ∈ B,

(L ◦ F)(b) = L(F(b)) = L(the city in which b is located)
= the country in which b is located.
A diagram of this function is shown in Figure 2.

Figure 2

2. Let g : Z → R be the function from part 2 of Example 5.1.3, which
was defined by the formula g(x) = 2x + 3. Let f : Z → Z be defined
2
by the formula f (n) = n − 3n + 1. Then g ◦ f : Z → R. For example,