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46 The Nuts and Bolts of Proof, Third Edition
is true (remember that this situation is the only one for which the statement
"If A, then B" is false). Indeed, to prove that a statement is false, it is enough
to prove that it is false in just one instance.
Examples cannot replace the proof that a statement is true in general,
because examples deal with special cases. A counterexample can prove that a
statement is false in general, because it exhibits one case in which the
statement is false.
Consider the statement "Every real number has a reciprocal." We can
think of milHons of numbers that do have a reciprocal. But the existence of
one number with no reciprocal (the number zero does not have a reciprocal)
makes the statement "Every real number has a reciprocal" false. What is true
is the statement "Every nonzero real number has a reciprocal."
Sometimes the existence of a counterexample can help us understand why
a statement is not true and whether a restriction of the hypothesis (or the
conclusion) can change it into a true statement.
The statement "The equahty
{a + hf:= a^ + b^
holds for all pairs of real numbers a and b in which at least one of the two
numbers is zero" is a true statement. (Prove it.)
The discovery of a counterexample can save the time and effort spent
trying to construct a proof, but sometimes even counterexamples are not
easy to find. Moreover, there is no sure way of knowing when to look for a
counterexample. If the best attempts at constructing a proof have failed, then
it might make sense to look for a counterexample. This search might be
difficult, but if it's successful then it proves that the statement is false. If it is
unsuccessful, it will provide examples that support the statement and might
give an insight into why the statement is true. And this might give new ideas
for the construction of the proof.
EXAMPLE 1. For all real numbers x > 0, x^>x^.
Discussion: It might be a good idea to graph the functions x^ and x^ to
compare them. Let:
A. The number x is a positive real number. (We can use all the properties
and operations of real numbers.)
B. x^>x^
Proof: Let us look for a counterexample. If x = 0.5, then x^ — 0.125 and
x^ = 0.25. Therefore, in this case, x^ <x^. So the statement is false. •
Note that the statement "For all real numbers x > 1, x^>x^" is true.