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

48 The Nuts and Bolts of Proof, Third Edition
10. Let/and g be two odd functions defined for all real numbers. Their
sum, /+ g, is an even function defined for all real numbers. (See
front material of the book for the definitions of even and odd
functions and /+ g.)
11. Let/and g be two odd functions defined for all real numbers. Then
their quotient function f/g is an even function defined for all real
numbers. (See front material of the book for the definitions of even
and odd functions.)
12. A six-digit palindrome number is divisible by 11.
13. The sum of two numbers is a rational number if and only if both
numbers are rational.
14. Let / be an odd function defined for all real numbers. The function
g(x) = {f{x)f is even. (See front material of the book for the
definitions of even and odd functions.)
15. Let/be a positive function defined for all real numbers. The function
g{x) = (f{x)y is always increasing. (See front material of the book for
the definitions of increasing function.)
MATHEMATICAL INDUCTION
In general, we use this kind of proof when we need to show that a certain
statement is true for an infinite collection of natural numbers, and direct
verification is impossible. We cannot simply check that the statement is true
for some of the numbers in the collection and then generalize the result to
the whole collection. Indeed, if we did this, we would just provide examples,
and, as already mentioned several times, examples are not proofs.
Consider the following claim: The inequality
n^ < 5n\
is true for all counting numbers w > 3.
How many numbers should we check? Is the claim true because the
inequality holds true for n = 3, 4, 5, 6, ..., 30? We cannot check directly all
counting numbers n > 3. Therefore, we must look for another way to prove
this kind of statement.
The technique of proving a statement by using mathematical induction
{complete induction) consists of the following three steps:
1. Prove that the statement is true for the smallest number included in the
statement to be proved (base case).
2. Assume that the statement is true for an arbitrary number in the
collection (inductive hypothesis).