Suppose ne Z. If n is odd, then n² is odd. Suppose ne Z. If n² is odd, then n is odd. Prove that 2 is irrational.
Suppose ne Z. If n is odd, then n² is odd. Suppose ne Z. If n² is odd, then n is odd. Prove that 2 is irrational.
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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![**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 \( \sqrt{2} \) 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 \( a^2 - 4b \neq 0 \).
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 \( 21a + 30b = 1 \).
11. There exist no integers \( a \) and \( b \) for which \( 18a + 6b + 1 = 0 \).
12. For every positive \( x \in \mathbb{Q} \), 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 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cd21fb3-e059-4805-be9c-60b9d1b235c0%2F81400b3f-6451-48ff-b43e-6cb673fd3220%2Fhvcwrek_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 \( \sqrt{2} \) 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 \( a^2 - 4b \neq 0 \).
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 \( 21a + 30b = 1 \).
11. There exist no integers \( a \) and \( b \) for which \( 18a + 6b + 1 = 0 \).
12. For every positive \( x \in \mathbb{Q} \), 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 \
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