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

22 The Nuts and Bolts of Proof, Third Edition
We can conclude from the preceding tables that the original statement is
only equivalent to its contrapositive. The converse and the inverse are
logically equivalent to each other, but not to the original statement.
PROOF BY CONTRAPOSITIVE (AKA PROOF BY
CONTRADICTION OR INDIRECT PROOF)
In some cases we cannot use the kind of direct, straightforward arguments
we have already seen; that is, we cannot deduce conclusion B directly
from hypothesis A. This might happen because assuming that A is true
does not seem to give us enough information to allow us to prove that B
is true. In other cases, direct verification of the conclusion B would be
too time consuming or impossible. Therefore, we must find another starting
point.
Because a statement is logically equivalent to its contrapositive, we can
try to work with the contrapositive. This gives us a different starting point
because we will start by assuming that B is false, and we will prove that this
implies that A is false, as the contrapositive of the original statement is "If
'not B,' then 'not A.'"
Let us consider an example that illustrates the use of this technique.
EXAMPLE 7. Let n be an integer number. If the number In-{-A is even,
then n is even.
Discussion: In this case, the hypothesis and the conclusion are clearly
distinguishable; therefore we can set:
A. The number 7n + 4 is even. (Implicit hypothesis: We can use the
properties of integer numbers and their operations.)
B. The number n is even.
By hypothesis, ln-\-A — 2k for some integer number k (see the section on
facts and properties of numbers at the front of the book). If we try to solve
for n explicitly, we will need to divide by 7, and it is not evident that the
result of the division will be an integer and will give information on the
parity of n. Therefore, we will try to prove the original statement by using its
contrapositive.
Proof: Assume that B is false, and that "not B" is true, and use this as the
new hypothesis. We will start by assuming that "n is not even." This means
that "n is odd." Thus,
n = 2t-\-\