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

40 The Nuts and Bolts of Proof, Third Edition
Discussion: By hypothesis a and b are two distinct real numbers, so we can
use all the properties of real numbers, such as the order (or trichotomy)
property, because the statements deal with comparison of numbers. The
order property states that, given any two real numbers, x and y, one of the
three following relations holds: x < y, or x > y, or x = y.
To prove that the three statements are equivalent we will need at least
three separate proofs. We will construct four proofs, according to the
following diagram:
Proof:
Part 1. If i, then ii; that is:
By hypothesis:
i o a
Hi
If the number b is larger than the number a,
then their average is larger than a.
a < b.
Because we want to obtain a + fo,we can add either a or b to both sides of
the inequality. Because the conclusion we want to reach deals with a, we
could try adding a. Thus, we obtain:
Dividing by 2 yields:
2a < a-\-b.
a-\-b
This proves that the conclusion holds true. So, the statement "If i, then ii"
is true.
Part 2. If ii, then i; that is:
If the average of a and fe, —-—, is larger than a,
then b is larger then a.
By hypothesis:
a + b
Then,
2a <
a^-b