Page 270 - HOW TO PROVE IT: A Structured Approach, Second Edition

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256 Functions
set of all cities with population at least one million, then B is a subset of C, and
the image of B under L would be the set

L(B) ={L(b) | b ∈ B}
={n ∈ N |∃b ∈ B(L(b) = n)}
={n ∈ N | there is some city with population at least
one million that is located in the country n}.

Thus, L(B) is the set of all countries that contain a city with population at least
one million. Now let A be the subset of N consisting of all countries in Africa.
Then the inverse image of A under L is the set

L −1 (A) ={c ∈ C | L(c) ∈ A}
={c ∈ C | the country in which c is located is in Africa}.
Thus, L −1 (A) is the set of all cities in African countries.
Let’s do one more example. Let f : R → R be defined by the formula
2
f (x) = x , and let X ={x ∈ R | 0 ≤ x < 2}. Then
2
f (X) ={ f (x) | x ∈ X}={x | 0 ≤ x < 2}.
Thus, f (X) is the set of all squares of real numbers between 0 and 2 (includ-
ing 0 but not 2). A moment’s reflection should convince you that this set is
{x ∈ R | 0 ≤ x < 4}. Now let’s let Y ={x ∈ R | 0 ≤ x < 4} and compute
f −1 (Y). According to the definition of inverse image,

f −1 (Y) ={x ∈ R | f (x) ∈ Y}
={x ∈ R | 0 ≤ f (x) < 4}
2
={x ∈ R | 0 ≤ x < 4}
={x ∈ R |− 2 < x < 2}.
By now you have had enough experience writing proofs that you should
be ready to put your proof-writing skills to work in answering mathematical
questions. Thus, most of this section will be devoted to a research project
in which you will discover for yourself the answers to basic mathematical
questions about images and inverse images. To get you started, we’ll work out
the answer to the first question.
Suppose f : A → B, and W and X are subsets of A. A natural question you
might ask is whether or not f (W ∩ X) must be the same as f (W) ∩ f (X). It
seems plausible that the answer is yes, so let’s see if we can prove it. Thus,
our goal will be to prove that f (W ∩ X) = f (W) ∩ f (X). Because this is an