91Ó°ÊÓ

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) Suppose \(n \in \mathbb{Z}\). If \(n^{2}\) is odd, then \(n\) is odd.

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) Suppose \(a, b \in \mathbb{R} .\) If \(a\) is rational and \(a b\) is irrational, then \(b\) is irrational.

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) There exist no integers \(a\) and \(b\) for which \(18 a+6 b=1\).

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) For every \(x \in[\pi / 2, \pi], \sin x-\cos x \geq 1\)

Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.) If \(a\) and \(b\) are positive real numbers, then \(a+b \geq 2 \sqrt{a b}\).

Prove the following statements using any method from Chapters 4,5 or 6 . The product of any five consecutive integers is divisible by 120 . (For example, the product of 3,4,5,6 and 7 is 2520 , and \(2520=120 \cdot 21\).