Please a detailed prove for number 4,8 and 10 and why that approach was chosen ch.iastate.edu/class/BookOfProof.pdfExercises for Chapter 7Prove the following statements. These exercises are cumulative, covering alltechniques addressed in Chapters 4-7.1. Suppose x e Z. Then x is even if and only if 3x+5 is odd.2. Suppose xe Z. Then x is odd if and only if 3x + 6 is odd.3. Given an integer a, then a³ + a² + a is even if and only if a is even.4. Given an integer a, then a² + 4a +5 is odd if and only if a is even.5. An integer a is odd if and only if a³ is odd.6. Suppose x,y e R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x.Then (x + y)² = x² + y² if and only if x = 0 or y = 0.7. Suppose x,y e R.8. Suppose a,b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b(mod 5).9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a.10. If a € Z, then a³ = a (mod 3).F411. Suppose a, b e Z. Prove that (a-3)62 is even if and only if a is odd or b is even.12. There exist a positive real number x for which x² <√√x.13. Suppose a, b e Z. If a + b is odd, then a² + b² is odd.14. Suppose a € Z. Then a² la if and only if a € (-1,0,1}.15. Suppose a, b € Z. Prove that a + b is even if and only if a and b have the sameparity.16. Suppose a, b e Z. If ab is odd, then a² + b² is even.17. There is a prime number between 90 and 100.18. There is a set X for which NE X and N≤X.19. If ne N, then 20 +2¹+22 +23+24+...+2 = 2n+1 -1.20. There exists an neN for which 11 (2-1).!!A-OF6CD A-..F7HIųF8(T)F9prt scF10home8

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4-7. 1. Suppose xe Z. Then x is even if and only if 3x+5 is odd. 2. Suppose xe Z. Then x is odd if and only if 3x+6 is odd. 3. Given an integer a, then a³ + a² + a is even if and only if a is even. 4. Given an integer a, then a² +4a +5 is odd if and only if a is even. 5. An integer a is odd if and only if a³ is odd. 6. Suppose x, y € R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x. 7. Suppose x, y € R. Then (x + y)² = x² + y² if and only if x = 0 or y = 0. 8. Suppose a, b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5). 9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a. 10. If a e Z, then a³ = a (mod 3). 11. Suppose a, b e Z. Prove that (a-3)62 is even if and only if a is odd or b is even. 12. There exist a positive real number x for which x² <√x. 13. Suppose a bf Z Ifath is odd then a² + h2 is odd

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4-7. 1. Suppose xe Z. Then x is even if and only if 3x+5 is odd. 2. Suppose xe Z. Then x is odd if and only if 3x+6 is odd. 3. Given an integer a, then a³ + a² + a is even if and only if a is even. 4. Given an integer a, then a² +4a +5 is odd if and only if a is even. 5. An integer a is odd if and only if a³ is odd. 6. Suppose x, y € R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x. 7. Suppose x, y € R. Then (x + y)² = x² + y² if and only if x = 0 or y = 0. 8. Suppose a, b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5). 9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a. 10. If a e Z, then a³ = a (mod 3). 11. Suppose a, b e Z. Prove that (a-3)62 is even if and only if a is odd or b is even. 12. There exist a positive real number x for which x² <√x. 13. Suppose a bf Z Ifath is odd then a² + h2 is odd

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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Question

Please a detailed prove for number 4,8 and 10 and why that approach was chosen

ch.iastate.edu/class/BookOfProof.pdf
Exercises for Chapter 7
Prove the following statements. These exercises are cumulative, covering all
techniques addressed in Chapters 4-7.
1. Suppose x e Z. Then x is even if and only if 3x+5 is odd.
2. Suppose xe Z. Then x is odd if and only if 3x + 6 is odd.
3. Given an integer a, then a³ + a² + a is even if and only if a is even.
4. Given an integer a, then a² + 4a +5 is odd if and only if a is even.
5. An integer a is odd if and only if a³ is odd.
6. Suppose x,y e R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x.
Then (x + y)² = x² + y² if and only if x = 0 or y = 0.
7. Suppose x,y e R.
8. Suppose a,b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b
(mod 5).
9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a.
10. If a € Z, then a³ = a (mod 3).
F4
11. Suppose a, b e Z. Prove that (a-3)62 is even if and only if a is odd or b is even.
12. There exist a positive real number x for which x² <√√x.
13. Suppose a, b e Z. If a + b is odd, then a² + b² is odd.
14. Suppose a € Z. Then a² la if and only if a € (-1,0,1}.
15. Suppose a, b € Z. Prove that a + b is even if and only if a and b have the same
parity.
16. Suppose a, b e Z. If ab is odd, then a² + b² is even.
17. There is a prime number between 90 and 100.
18. There is a set X for which NE X and N≤X.
19. If ne N, then 20 +2¹+22 +23+24+...+2 = 2n+1 -1.
20. There exists an neN for which 11 (2-1).
!!
A
-O
F6
CD A
-..
F7
HI
ų
F8
(T)
F9
prt sc
F10
home
8

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