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

Special Kinds of Theorems 47
EXAMPLE 2. If a positive integer number is divisible by a prime number,
then it is not prime.
Proof: The statement is false. Consider the prime number 7. It is
a positive integer number and it is divisible by the prime number 7 (indeed
7/7 = 1). So, it satisfies the hypothesis, but 7 is a prime number. Thus, the
conclusion is false. •
The statement "If a positive integer number is divisible by a prime
number and the quotient of the division is not 1, then it is not prime" is true.
EXAMPLE 3. If an integer is a multiple of 10 and 15, then it is a multiple
of 150.
Proof. The statement is false. Just consider the least common multiple
of 10 and 15, namely 30. This number is a multiple of 10 and 15, but it is
not a multiple of 150. •
The statement "If an integer is a multiple of 10 and 15, then it is a multiple
of 30" is true.
EXERCISES
Use counterexamples to prove that the following statements are false.
1. Let/be an increasing function and gf be a decreasing function. Then
the function /+ g is constant. (See front material of the book for the
definitions of nonincreasing function and /+ g.)
2. If t is an angle in the first quadrant, then 2 sin t = sin 2t.
3. Consider the polynomial P(x) = —x^ + 2x — 3/4. If j = P(^X then y
is always negative.
4. The reciprocal of a real number x > 1 is a number y such that
0<3;<1.
5. The number 2" + 1 is prime for all counting numbers n.
6. Let /, g, and h be three functions defined for all real numbers. If
fog=foh, then g = h.
Discuss the truth of the following statements; that is, prove those that are
true and provide counterexamples for those that are false.
7. The sum of any five consecutive integers is divisible by 5.
8. If/(x) = x^ and g{x) = x'^, then/(x) < g(x) for all real numbers x; > 0.
9. The sum of four consecutive counting numbers is divisible by 4 (see
exercise 7).