Book of Proof - Amazon S3

CHAPTER 9
Disproof
E
ver since Chapter 4 we have dealt with one major theme: Given a
statement, prove that is it true. In every example and exercise we
were handed a true statement and charged with the task of proving it.
Have you ever wondered what would happen if you were given a false
statement to prove? The answer is that no (correct) proof would be possible,
for if it were, the statement would be true, not false.
But how would you convince someone that a statement is false? The
mere fact that you could not produce a proof does not automatically mean
the statement is false, for you know (perhaps all too well) that proofs
can be difficult to construct. It turns out that there is a very simple and
utterly convincing procedure that proves a statement is false. The process
of carrying out this procedure is called disproof. Thus, this chapter is
concerned with disproving statements.
Before describing the new method, we will set the stage with some
relevant background information. First, we point out that mathematical
statements can be divided into three categories, described below.
One category consists of all those statements that have been proved to be
true. For the most part we regard these statements as significant enough
to be designated with special names such as “theorem,” “proposition,”
“lemma” and “corollary.” Some examples of statements in this category are
listed in the left-hand box in the diagram on the following page. There are
also some wholly uninteresting statements (such as 2 = 2) in this category,
and although we acknowledge their existence we certainly do not dignify
them with terms such as “theorem” or “proposition.”
At the other extreme is a category consisting of statements that are
known to be false. Examples are listed in the box on the right. Since
mathematicians are not very interested in them, these types of statements
do not get any special names, other than the blanket term “false statement.”
But there is a third (and quite interesting) category between these
two extremes. It consists of statements whose truth or falsity has not
been determined. Examples include things like “Every perfect number