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

Special Kinds of Theorems 61
EXERCISES
Prove the following statements.
1. There exists a function whose domain consists of all the real numbers
and whose range is in the interval [0,1].
2. There is a counting number n such that 2" + 7" is a prime number.
3. Let a be an irrational number. Then there exists an irrational number b
such that ab is an integer.
4. There is a second degree polynomial P such that P(0) = —1 and
P{-1) = 2.
5. There exist two rational numbers a and b such that af^ is a positive
integer and b^ is a negative integer.
6. If P{x) = a„x" + a„_ix"~^ H h aix + ao is a polynomial of degree n,
with n odd, then the equation P(x) = 0 has at least one real solution. (*)
7. Let a and b be two rational numbers, with a <b. Then there exist at
least three rational numbers between a and b.
8. There exists an integer number k such that 2^ > 4^.
UNIQUENESS THEOREMS
This kind of theorem states that an object having some required properties,
and whose existence has already been established, is unique. In order to
prove that this is true, we have to prove that no other object satisfies the
properties Hsted. Direct and exphcit checking is usually impossible, because
we might be deahng with infinite collections of objects. Therefore, we need to
use a different approach. Usually this kind of theorem is proved in one of the
three following ways:
L What would happen if the object with the required properties is not
unique? To deny that something is unique means to assume that there
is at least one more object with the same properties. So, assume that
there are two objects satisfying the given properties, and then prove
that they coincide.
2. If the object has been exphcitly constructed using an algorithm
(a procedure), we might be able to use the fact that every step of the
algorithm could only be performed in a unique way.
3. Especially when the explicit construction of the object is not possible,
we might be able to find a general argument that guarantees the
uniqueness of the object with the required properties.