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

Special Kinds of Theorems 89
where p / 0, and n and p are integer numbers.
As X is rational, we can write x = a/b with b^O, where a and b are integer
numbers. Thus, we have:
a/b-\-y = n/p
If we solve this equation for y, we obtain:
y = n/p — a/b = (nb — ap)/pb
where pb^O because p / 0 and b^O.
The numbers nb - ap and pb are integers because n, p, a, and b are integer
numbers.
This information allows us to conclude that y is indeed a rational number.
As we have proved the contrapositive of the original statement to be true,
the original statement is also true. •
EXAMPLE 7. Let a be an even number, with \a\ > 16. Then either a> 18
or a<-18.
Discussion: In spite of its apparent simplicity, this statement has composite
hypotheses and conclusions. Indeed, it is of the form "If A and B, then C or
D," where:
A. The number a is even.
{Implicit hypothesis: We can use properties and operations of integer
numbers.)
The fact that the number is called a is irrelevant.
B. |a|>16.
C. a > 18.
D. a<-18.
Moreover, B is a composite statement. Indeed, B can be written as Bi or B2,
with
Bi : a > 16
B2 : a < -16.
and
Thus, the original statement can be rewritten as:
If (a is even and a> 16) or (a is even and a < —16),
then either a > 18 or a < -18.
The presence of an "or" in the hypothesis suggests the construction of a
proof by cases.