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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1. Suppose ne Z. If n is odd, then n² is odd.
2. Suppose ne Z. If n² is odd, then n is odd.
3. Prove that V2 is irrational.
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.)
F4
4. Prove that √6 is irrational.
5. Prove that √3 is irrational.
6. If a, b e Z, then a²-4b-20.
7. If a, b e Z, then a²-4b-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 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) = Ø.
15. If b € Z 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.
H
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CD A
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Transcribed Image Text:F3
edu/class/BookUfProot.pat
1. Suppose ne Z. If n is odd, then n² is odd.
2. Suppose ne Z. If n² is odd, then n is odd.
3. Prove that V2 is irrational.
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.)
F4
4. Prove that √6 is irrational.
5. Prove that √3 is irrational.
6. If a, b e Z, then a²-4b-20.
7. If a, b e Z, then a²-4b-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 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) = Ø.
15. If b € Z 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.
H
O
F5
CD A
F6
H
F8
T
F9
prt sc
F10
hor
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