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

Special Kinds of Theorems 81
This implication is trivially true. Indeed, if |fl| =0, then |a| is smaller than
any positive number.
Part 2. If \a\ < s for every real number ^ > 0, then \a\ =0.
We will prove this statement by using its contrapositive.
Let us assume that |a| T^^O. Does this imply that there exists at least one
positive real number ^o such that |a| is not smaller that ^o?
Consider the number ^o = 1^1/2. Then 0<£o< Ml-
Because the contrapositive of the original statement is true, the original
statement is true as well. •
EXERCISES
Prove the following statements:
1. Let X and y be two real numbers.
Then {x — yf + (x - y)^ = 0 if and only ii x = y.
2. Let X and y be two real numbers. The two sequences {x"}^2 ^^^
{y"}^2 ^^^ equal if and only ii x = y.
3. Let a, b, and c be three counting numbers. If a divides b, b divides c,
and c divides a, then a = b = c.
4. Let a, b, and c be three counting numbers. Then GCD{ac, be) =
cGCD{a,by
5. Let a and b be two relatively prime integers. If there exists an m such
that (a/b)"^ is an integer, then b=l.
COMPOSITE STATEMENTS
The hypothesis and conclusion of a theorem might be composite
statements. Because of the more complicated structure of this kind of
statement, we have to pay very close attention. After analyzing a composite
statement, we can check if it is possible to break it down into simpler parts,
which can then be proved by using any of the principles and techniques
already seen. Other times we will replace the original statement with another
logically equivalent to it, but easier to handle.
MULTIPLE HYPOTHESES
Multiple hypotheses statements are statements for which the hypotheses
are composite statements, such as "If A and B, then C" and "If A or B,
then C."