Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.2 CONCLUSIONS INVOLVING V AND +, BUT NOT 3 lm

definition of C, we know that (-x, f(-x)) E C. Since C is the graph of a

function, and so can have no more than one y value corresponding to

any given x value, and since both f(x) and f(-x) correspond to -x,

we must have f(x) = f (- x), as desired. 0

The proof in Example 12 is perhaps the most difficult one we have
presented thus far, although the result itself is exceedingly plausible. Some-
times a result can seem so "obvious" that we fail to notice that the defini-
tions involved are distinct and that there indeed is something to be proved.
A formal proof, in such a case, is merely a rigorous explanation of why the
result is obvious. Since intuition can mislead, it is important, at all levels
of mathematics, to know how to write a formal proof, if pressed, of all
mathematical statements that we claim are true, even though we do not, in
practice, write out every such proof.
In this article we have stated that, for many proofs, the correct ap-
proach-the proper "setting-up" of the argument-is a very large part of
the "battle." Although this is true, there are, of course, many proofs that
are more complicated and require more than just the proper technical ap-
proach, both in terms of prior knowledge of relevant mathematics and in
terms of facility with further proof techniques. The .= part of Example 12
demonstrated both needs. At the very end of the proof, we had to call on
some general knowledge about what a function is (i.e., "no x value has two
distinct corresponding y values"). Just before that, when we noted that the
ordered pair ( - x, f ( - x)) E C, we were implicitly using a technique of proof
known as specialization. We knew that C consisted of all ordered pairs
of the form (a, f(a)), where a ranges over all real numbers. Thus, in par-
ticular, the ordered pair (- x, f (- x)) must be in C, where x is the arbitrary
real number whose value we fixed at the start of the proof. Specialization
is one of two very useful techniques we will focus on in the next article
(division into cases being the other). At the end of that article, we will be
able to handle a wider variety of problems calling for the derivation of con-
clusions involving V and + than in the exercises that follow.

Exercises -

1. Let A and B be arbitrary sets. Prove:
   (a) A n B = A if and only if A c B
   (b) If A u B = B, then A c B (The converse is also true. It will appear as an ex-
   ercise at the end of the next article.)
   (c) If C is a nonempty set such that A x C = B x C, then A = B (recall Exam-
   ple 9. Note also the connection between this exercise and Exercise 5, Article
   1.3.).