Chapter 6

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 \(n\) is odd, then \(n^{2}\) 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 \(n \in \mathbb{Z}\). 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.) Prove that \(\sqrt[3]{2}\) 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.) Prove that \(\sqrt{6}\) 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.) Prove that \(\sqrt{3}\) 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.) Suppose \(a, b, c \in \mathbb{Z} .\) If \(a^{2}+b^{2}=c^{2},\) then \(a\) or \(b\) is even.

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 \(21 a+30 b=1\).

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.) 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 positive \(x \in \mathbb{Q},\) there is a positive \(y \in \mathbb{Q}\) for which \(y

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 sets, then \(A \cap(B-A)=\varnothing\).

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 \(b \in \mathbb{Z}\) and \(b \nmid k\) for every \(k \in \mathbb{N},\) then \(b=0\).

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 \(a+b \geq 2 \sqrt{a b}\).

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 \(n \in \mathbb{Z}, 4 \nmid\left(n^{2}+2\right)\)

For every \(n \in \mathbb{Z}, 4 \nmid\left(n^{2}+2\right)\). Suppose \(a, b \in \mathbb{Z}\). If \(4 \mid\left(a^{2}+b^{2}\right),\) then \(a\) and \(b\) are not both odd.

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 \(2520=120 \cdot 21\).

Prove the following statements using any method from Chapters 4,5 or 6 . Explain why \(x^{2}+y^{2}-3=0\) not having any rational solutions (Exercise 20 ) implies \(x^{2}+y^{2}-3^{k}=0\) has no rational solutions for \(k\) an odd, positive integer.

Prove the following statements using any method from Chapters 4,5 or 6 . Use the above result to prove that \(\sqrt{3^{k}}\) is irrational for all odd, positive \(k\).

Prove the following statements using any method from Chapters 4,5 or 6 . The number \(\log _{2} 3\) is irrational.