8. Suppose a, b, c e Z. If a² + b² = c², then a or b is even. 9. Suppose a, b E R. If a is rational and ab is irrational, then b is irrational. 10. There exist no integers a and b for which 21a + 30b = 1. 11. There exist no integers a and b for which 18a +6b = 1. 12. For every positive x e Q, there is a positive ye Q for which y < x. 13. For every x € [л/2, л], sinx-cosx ≥ 1. 14. If A and B are sets, then An (B-A) = 0. 15. If beZ and błk for every k EN, then b=0. 16. If a and b are positive real numbers, then a + b ≥ 2√ab.
8. Suppose a, b, c e Z. If a² + b² = c², then a or b is even. 9. Suppose a, b E R. If a is rational and ab is irrational, then b is irrational. 10. There exist no integers a and b for which 21a + 30b = 1. 11. There exist no integers a and b for which 18a +6b = 1. 12. For every positive x e Q, there is a positive ye Q for which y < x. 13. For every x € [л/2, л], sinx-cosx ≥ 1. 14. If A and B are sets, then An (B-A) = 0. 15. If beZ and błk for every k EN, then b=0. 16. If a and b are positive real numbers, then a + b ≥ 2√ab.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please I need a detailed prove for 8,16 and 12
![# Exercises for Chapter 6
## A. 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.)
1. Suppose \( n \in \mathbb{Z} \). If \( n \) is odd, then \( n^2 \) is odd.
2. Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is odd, then \( n \) is odd.
3. Prove that \(\sqrt[3]{2}\) is irrational.
4. Prove that \(\sqrt{6}\) is irrational.
5. Prove that \(\sqrt{3}\) is irrational.
6. If \( a, b \in \mathbb{Z} \), then \( a^2 - 4b \neq 0 \).
7. If \( a, b \in \mathbb{Z} \), then \( a^2 - 4b - 3 \neq 0 \).
8. Suppose \( a, b, c \in \mathbb{Z} \). If \( a^2 + b^2 = c^2 \), then \( a \) or \( b \) is even.
9. Suppose \( a, b \in \mathbb{R} \). If \( a \) is rational and \( ab \) is irrational, then \( b \) is irrational.
10. There exist no integers \( a \) and \( b \) for which \( 10a + 30b = 1 \).
11. There exist no integers \( a \) and \( b \) for which \( 18a + 6b = 1 \).
12. For every positive \( x \in \mathbb{R} \), there is a positive \( y \in \mathbb{Q} \) for which \( y < x \).
13. For every \( x \in [\pi/2, \pi] \), \(\sin x - \cos x \geq 1\).
14. If \( A \) and \( B \) are sets, then \( A \cap (B - A) = \emptyset \).
15. If \( b \in \mathbb{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cd21fb3-e059-4805-be9c-60b9d1b235c0%2F5a936f41-77d7-4b00-a4a5-013d84c03604%2Fmsgloma_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Exercises for Chapter 6
## A. 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.)
1. Suppose \( n \in \mathbb{Z} \). If \( n \) is odd, then \( n^2 \) is odd.
2. Suppose \( n \in \mathbb{Z} \). If \( n^2 \) is odd, then \( n \) is odd.
3. Prove that \(\sqrt[3]{2}\) is irrational.
4. Prove that \(\sqrt{6}\) is irrational.
5. Prove that \(\sqrt{3}\) is irrational.
6. If \( a, b \in \mathbb{Z} \), then \( a^2 - 4b \neq 0 \).
7. If \( a, b \in \mathbb{Z} \), then \( a^2 - 4b - 3 \neq 0 \).
8. Suppose \( a, b, c \in \mathbb{Z} \). If \( a^2 + b^2 = c^2 \), then \( a \) or \( b \) is even.
9. Suppose \( a, b \in \mathbb{R} \). If \( a \) is rational and \( ab \) is irrational, then \( b \) is irrational.
10. There exist no integers \( a \) and \( b \) for which \( 10a + 30b = 1 \).
11. There exist no integers \( a \) and \( b \) for which \( 18a + 6b = 1 \).
12. For every positive \( x \in \mathbb{R} \), there is a positive \( y \in \mathbb{Q} \) for which \( y < x \).
13. For every \( x \in [\pi/2, \pi] \), \(\sin x - \cos x \geq 1\).
14. If \( A \) and \( B \) are sets, then \( A \cap (B - A) = \emptyset \).
15. If \( b \in \mathbb{
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