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90 The Nuts and Bolts of Proof, Third Edition
Proof:
Case 1. We will prove the statement:
If a is an even number and a > 16,
then either a > 18 or a < —18.
As a is even and larger than 16, then it must be at least 18. Thus, a > 18, and
the conclusion is true.
Case 2. We will prove the statement:
If a is an even number and a < —16,
then either a > 18 or a < -18.
As a is even and smaller than -16, then it cannot be —17, so it must be
at most —18. Therefore, a< -18, and the conclusion is true. •
EXERCISES
Prove the following statements.
1. If x^ = y^ where x > 0 and y > 0, then x = y.
2. If a function / is even and odd, then /(x) = 0 for all x in the domain
of the function.
(See the front material of the book for the definitions of even and odd
functions.)
3. If n is a positive multiple of 3, then either n is odd or it is a multiple of 6.
4. If X and y are two real numbers such that x^ = y^, then either x = >;
or X = —y,
5. Let A and B be two subsets of the same set [7. Define the difference
set A - B as:
A~B={aeA\a^B\.
If ^ - 5 is empty, then either A is empty ox AC.B.
6. Let A and B be two sets. If either yl = 0 or .4CB, then A\JB = B.
7. Fill in all the details and outline the following proof of the rational
zero theorem:
Let z be a rational zero of the polynomial:
P(x) = a„x" + a„_ix"~^ H
Va^