Chapter 1 What is a Proof?24
Problems for Section 1.8
Practice Problems
Problem 1.8.
Prove that for anyn > 0, ifanis even, thenais even.
Hint:Contradiction.
Problem 1.9.
Prove that ifabDn, then eitheraorbmust be
p
n, wherea;b, andnare
nonnegative real numbers.Hint:by contradiction, Section 1.8.
Problem 1.10.
Letnbe a nonnegative integer.
(a)Explain why ifn^2 is even—that is, a multiple of 2—thennis even.(b)Explain why ifn^2 is a multiple of 3, thennmust be a multiple of 3.Problem 1.11.
Give an example of two distinct positive integersm;nsuch thatn^2 is a multiple of
m, butnis not a multiple ofm. How about havingmbe less thann?
Class Problems
Problem 1.12.
How far can you generalize the proof of Theorem 1.8.1 that
p
2 is irrational? For
example, how about
p
3?Problem 1.13.
Prove that log 46 is irrational.
Problem 1.14.
Here is a different proof that
p
2 is irrational, taken from the American Mathemat-
ical Monthly, v.116, #1, Jan. 2009, p.69:
Proof. Suppose for the sake of contradiction that
p
2 is rational, and choose the