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

88 The Nuts and Bolts of Proof, Third Edition
{Implicit hypothesis: We can use all properties of counting and prime,
non-prime numbers, divisibility, and the properties of square roots.)
B or C. Then either p < y/n ov q< v^.
Proof: We will start by assuming that:
n = pq
where \ <p <n and I < q<n, and p>y/n.
Multiplying this last inequality by q yields:
that is,
qp > q^/n,
n > q\fn.
This implies y/n> q. Thus, it is true that yfn>:q.
(Note that '\ip — q, then p = q = y/n.) •
The result stated in Example 5 is used to improve the speed of the search
for possible prime factors of numbers.
EXAMPLE 6. If x is a rational number and y is an irrational number, their
sum, X + y, is an irrational number.
Discussion: We will prove the contrapositive of the given statement; that is,
the statement "If the sum x + y is a rational number, then either x is
irrational or y is rational." Thus, let:
A. The sum x + }^ is a rational number.
{Implicit hypothesis: As rational and irrational numbers are real numbers,
we can use all the properties of real numbers and their operations.)
The fact that the numbers are called x and y is irrelevant. We can use any
two symbols. We will keep using x and y to be consistent with the original
statement.
B. The number x is irrational.
C. The number y is rational.
Therefore, we plan to prove the equivalent statement "If A and 'not B,'
then C."
Proof: Assume that the number x + 3; is rational and so is the number x.
Therefore, using the definition of rational numbers, we can write:
x-{-y=:n/p