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

Introduction and Basic Terminology 23
for some integer number t. Using this information to calculate the number
ln-\-4 yields:
7n + 4 = 7(2t + 1) + 4 = 2(7t + 5) + 1.
The number 5 = 7t + 5 is an integer because 7, t, and 5 are integers; therefore,
we can write:
7n + 4 = 2s + 1
and conclude that 7n 4- 4 is an odd number ("not A"). This means that the
statement "If 'not B,' then 'not A'" is true, and its contrapositive {i.e., the
original statement) is also true. •
EXAMPLE 8. Let m and n be two integers. If they are consecutive, then
their sum, m + n, is an odd number.
Discussion: In this case we can set:
A. The numbers m and n are consecutive
{i.e., if m is the larger of the two, then m-n=l).
{Implicit hypothesis: We can use the properties of integer numbers
and their operations.)
B. The sum m + n is an odd number.
This statement can be proved either directly (this proof is left as an exercise)
or by using its contrapositive.
Proof: Assume that B is false and "not B" is true, and use this as the new
hypothesis. We will start by assuming that the number m + n is not odd.
Then, m + n is even and there exists some integer number k such that
m-\-n = 2k. This implies that m = 2k — n, thus:
m — n = {2k — n) — n = 2{k — n).
As the number k — n is an integer, we have proved that the difference
between m and n is an even number; therefore, it cannot equal 1, and m
and n are not consecutive numbers. This means that the statement "If'not B,'
then 'not A'" is true, and its contrapositive {i.e., the original statement) is
also true. •
EXAMPLE 9. There are infinitely many prime numbers.
Discussion: We need to analyze the statement to find the hypothesis and the
conclusion because they are not clearly distinguishable. The point is that we