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36 The Nuts and Bolts of Proof, Third Edition
Notice that the statements "If A, then B" and "If B, then A" are converses
of each other.
EXAMPLE 1. A nonzero real number is positive if and only if its
reciprocal is positive.
Proof: Consider:
A. A real number a is positive.
B. The reciprocal of a, denoted as a~\ is positive.
Parti. If A, then B.
The fact that the number a is positive is sufficient to imply that its reciprocal
is positive. By definition of a reciprocal:
ax a~^ = 1.
So, the number ax a~^ is positive.
By the properties of operations of real numbers, the product of two
numbers is positive only if the two numbers are either both positive or both
negative. Because by hypothesis a is positive, it follows that a~^ is positive.
Part 2. If B, then A.
The fact that the number a is positive is necessary to imply that its reciprocal
is positive. By definition of reciprocal
a X a~^ = 1.
So, the number ax a~^ is positive.
By the properties of operations of real numbers, the product of two
numbers is positive only if the two numbers are either both positive or
both negative. By hypothesis, a~^ is positive, thus it follows that a is
positive. •
It is easy to see that the two parts of the proof in Example 1 are very
similar. Thus, after making sure that no details have been overlooked, we
can edit and streamline the proof. The following is an example of how the
proof can be condensed.
Let a X a~^ = 1. The product of two numbers is positive if and only if the
two numbers are either both positive or both negative. Thus, if a is positive,
so is fl~\ and, conversely, if a~^ is positive, so is a.
EXAMPLE 2. A counting number is odd if and only if its square is odd.
Proof: Let n represent a generic counting number. Then we can set:
A. The number n is odd.
B. The number n^ is odd.