Page 256 - HOW TO PROVE IT: A Structured Approach, Second Edition

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242 Functions
Commentary

1. AsinExample5.2.4,weprovethat g ◦ f isone-to-onebyprovingthat∀a 1 ∈
A∀a 2 ∈ A((g ◦ f )(a 1 ) = (g ◦ f )(a 2 ) → a 1 = a 2 ). Thus, we let a 1 and a 2
be arbitrary elements of A, assume that (g ◦ f )(a 1 ) = (g ◦ f )(a 2 ), which
means g( f (a 1 )) = g( f (a 2 )), and then prove that a 1 = a 2 . The next sentence
of the proof says that the assumption that g is one-to-one is being used, but
it might not be clear how it is being used. To understand this step, let’s
write out what it means to say that g is one-to-one. As we observed before,
rather than using the original definition, which is a negative statement,
we are probably better off using the equivalent positive statement ∀b 1 ∈
B∀b 2 ∈ B(g(b 1 ) = g(b 2 ) → b 1 = b 2 ). The natural way to use a given of
this form is to plug something in for b 1 and b 2 . Plugging in f (a 1 ) and
f (a 2 ), we get g( f (a 1 )) = g( f (a 2 )) → f (a 1 ) = f (a 2 ), and since we know
g( f (a 1 )) = g( f (a 2 )), it follows by modus ponens that f (a 1 ) = f (a 2 ). None
of this was explained in the proof; readers of the proof are expected to work it
out for themselves. Make sure you understand how, using similar reasoning,
you can get from f (a 1 ) = f (a 2 )to a 1 = a 2 by applying the fact that f is
one-to-one.
2. After the assumption that f and g are both onto, the form of the rest of the
proof is entirely guided by the logical form of the goal of proving that g ◦ f
is onto. Because this means ∀c ∈ C∃a ∈ A((g ◦ f )(a) = c), we let c be an
arbitrary element of C and then find some a ∈ A for which we can prove
(g ◦ f )(a) = c.

Functions that are both one-to-one and onto are particularly important in
mathematics.Suchfunctionsaresometimescalledone-to-onecorrespondences
or bijections. Figure 2(b) shows an example of a one-to-one correspondence.
Notice in the figure that both A and B have four elements. In fact, you should be
able to convince yourself that if there is a one-to-one correspondence between
two finite sets, then the sets must have the same number of elements. This is
one of the reasons why one-to-one correspondences are so important. We will
discuss one-to-one correspondences between infinite sets in Chapter 7.
Here’s another example of a one-to-one correspondence. Suppose A is the
set of all members of the audience at a sold-out concert and S is the set of all
seats in the concert hall. Let f : A → S be the function defined by the rule

f (a) = the seat in which a is sitting.
Because different people would not be sitting in the same seat, f is one-to-one.
Because the concert is sold out, every seat is taken, so f is onto. Thus, f is a