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

Introduction and Basic Terminology 13
We reached the conclusion that was part of the original statement! We
seem to be on the right track. Can we rewrite the proof in a precise and easy
to follow way? Let us try!
Proof: Let a and b be two odd numbers. As the numbers are odd, it is
possible write:
a = 2t-\-l and b = 2s + 1
where t and s are two integers. Therefore,
a + fc = (2t + 1) + (25 + 1) = 2t + 2s + 2 = 2(t + s + 1)
The number /? = t + s + 1 is an integer because t and s are integers. Thus,
a-\-b = 2p
where p is an integer.
This implies that a + ft is an even number. Because this is the conclusion
in the original statement, the proof is complete. •
Let us look back briefly at how this proof relates to the considerations
presented at the beginning of this section. We have worked under the
assumption that part A of the statement is true. We have shown that part B
holds true, and we have done this using a general way of thinking, not by
using specific examples (more about this later). Therefore, it is true that A
implies B. Now, let us consider another statement.
EXAMPLE 2. If the x- and the y-intercepts of a line that does not pass
through the point (0,0) have rational coordinates, then the slope of the line is
a rational number.
Discussion: Let us separate the hypothesis and conclusion:
A. Consider a hne in the Cartesian plane such that its x- and y-
intercept have rational coordinates, and neither one of them is the
point (0,0).
B. The slope of the hne described in the hypothesis is a rational number.
Implicit hypothesis: We need to know the structure of the Cartesian plane,
how to find the x- and the y-intercepts of a line, how to find the slope of a
hne, and how to use the properties and operations of rational numbers.
The hypothesis mentions two special points on the hne—namely its x-
and 3;-intercepts. In general, if we know the coordinates of any two points on
a line, we can use them to calculate the slope of the hne. Indeed, if A{xi,yi)