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.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter1: Line And Angle Relationships
Section1.4: Relationships: Perpendicular Lines
Problem 12E: In Exercise 11 and 12, provide the missing statements and reasons. Given: 12;34 s 2 and 3 are...
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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 e Z. If n is odd, then n² is odd.
2. Suppose ne Z. If n² is odd, then n is odd.
3. Prove that √2 is irrational.
4. Prove that √6 is irrasional.
5. Prove that √3 is irrational.
6. If a,b € Z, then a²-4b-20.
7. If a, b e Z, then a²-46-30.
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 y EQ 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.
17. For every ne Z, 4 (n²+2).
18. Suppose a, b e Z. If 4 | (a² + b²), then a and b are not both odd.
B. Prove the following statements using any method from Chapters 4, 5 or 6.
DII
F4
€0
FS
F6
F7
CDA Y
H
8
F8
F9
prt sc
F10
he](/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 e Z. If n is odd, then n² is odd.
2. Suppose ne Z. If n² is odd, then n is odd.
3. Prove that √2 is irrational.
4. Prove that √6 is irrasional.
5. Prove that √3 is irrational.
6. If a,b € Z, then a²-4b-20.
7. If a, b e Z, then a²-46-30.
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 y EQ 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.
17. For every ne Z, 4 (n²+2).
18. Suppose a, b e Z. If 4 | (a² + b²), then a and b are not both odd.
B. Prove the following statements using any method from Chapters 4, 5 or 6.
DII
F4
€0
FS
F6
F7
CDA Y
H
8
F8
F9
prt sc
F10
he
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