epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Introduction and Basic Terminology 29
we can start from the assumption that there exists a positive number x such
that:
2
X < X .
We can rewrite the previous inequaHty as:
X — x^ <0.
The expression on the left-hand side of the inequahty can be factored, and
the inequahty becomes:
x(l - x) < 0.
The product of two real numbers is negative if and only if the numbers have
opposite sign. Because, by hypothesis, x > 0, we must conclude that:
1 - X < 0; that is, 1 < x.
Thus, we proved that "not A" is true. Because the contrapositive of the
original statement is true, the original statement is true as well. •
In Example 11, the direct proof is shorter and simpler, but sometimes it is
nice to know that there is another way to achieve a goal.
As already mentioned, one has to use caution identifying the hypothesis
and conclusion to construct and use the contrapositive of a statement.
Sometimes a theorem might include "overarching" hypotheses that are just
used to define the general setting in which the statement "if A, then B"
should be considered. This kind of hypothesis will not be changed. For
example, consider the following statement: "Let x, y, and z be counting
numbers. If xy is not a multiple of z, then x is not a multiple of z and y is
not a multiple of z." When we construct its contrapositive, we do not deny
the fact that x, y, and z are counting numbers. We will instead consider
the statement: "Let x, y, and z be counting numbers. If x is a multiple of z or
y is a multiple of z, then xy is a multiple of z."
Denying a statement that contains more than one quantifier can be
difficult at first. Some of these statements are found in calculus and real
analysis, in the definitions of limits and continuity. Let's examine one of
them:
A. The real number L is said to be the limit of the function /(x) at
the point c if for every s>0 there exists a 5 > 0 such that if 0|x - c| < 5,
then |/(x) - L\<£.