Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

1.1. Statements and Conditional Statements 9
Progress Check 1.4, you were asked if you thought the following conditional
statement was true or false.
If n is a positive integer, then n 2
n C 41 is a prime number.
Perhaps for all of the values you tried for n, n 2 n C 41 turned out to be
a prime number. However, if we try n D 41, weget
n 2 n C 41 D 41 2 41 C 41
n 2 n C 41 D 41 2 :
So in the case where n D 41, the hypothesis is true (41 is a positive integer)
and the conclusion is false 41 2 is not prime . Therefore, 41 is a counterexample
for this conjecture and the conditional statement
“If n is a positive integer, then n 2
n C 41 is a prime number”
is false. There are other counterexamples (such as n D 42, n D 45, and
n D 50), but only one counterexample is needed to prove that the statement
is false.
3. Although one example can be used to prove that a conditional statement is
false, in most cases, we cannot use examples to prove that a conditional
statement is true. For example, in Progress Check 1.4, we substituted values
for x for the conditional statement “If x is a positive real number, then
x 2 C 8x is a positive real number.” For every positive real number used
for x, wesawthatx 2 C 8x was positive. However, this does not prove the
conditional statement to be true because it is impossible to substitute every
positive real number for x. So, although we may believe this statement is
true, to be able to conclude it is true, we need to write a mathematical proof.
Methods of proof will be discussed in Section 1.2 and Chapter 3.
Progress Check 1.5 (Working with a Conditional Statement)
Sometimes, we must be aware of conventions that are being used. In most calculus
texts, the convention is that any function has a domain and a range that are subsets
of the real numbers. In addition, when we say something like “the function f is
differentiable at a”, it is understood that a is a real number. With these conventions,
the following statement is a true statement, which is proven in many calculus texts.
If the function f is differentiable at a, then the function f is continuous at a.