Number 8 Constructive Versus Non-Constructive ProofsThis existence proof has inside of it a separate proof (by contradiction)in other books and articles, as well as to the possibility of crafting suchBe alert to constructive and non-constructive proofs as you read proofsthat log2 9 is irrational. Such combinations of proof techniques are, of course,tements155uctive.corem;ing it.typical.existproofs of your own. iomsatoonal,Exercises for Chapter 7Drove the following statements. These exercises are cumulative, covering allearwetechniques addressed in Chapters 4–7.1. Suppose x € Z. Then x is even if and only if 3x +5 is odd.2. Suppose x € 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 +x² y = y² +xy if and only if y = x² or y = -x.8, Suppose a,b e Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b(mod 5).7. Suppose x,y e R. Then (x + y)² = x² + y² if and only if x = 0 or y = 0.9. Suppose a e 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,be Z. Prove that (a - 3)62 is even if and only if a is odd or b is even.13. Suppose a,b e Z. If a + b is odd, then a² +b² is odd.14. Suppose a e Z. Then a² | a if and only if a e{-1,0,1}.12, There exists a positive real number x for which x2

Question

Number 8 Constructive Versus Non-Constructive ProofsThis existence proof has inside of it a separate proof (by contradiction)in other books and articles, as well as to the possibility of crafting suchBe alert to constructive and non-constructive proofs as you read proofsthat log2 9 is irrational. Such combinations of proof techniques are, of course,tements155uctive.corem;ing it.typical.existproofs of your own. iomsatoonal,Exercises for Chapter 7Drove the following statements. These exercises are cumulative, covering allearwetechniques addressed in Chapters 4–7.1. Suppose x € Z. Then x is even if and only if 3x +5 is odd.2. Suppose x € 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 +x² y = y² +xy if and only if y = x² or y = -x.8, Suppose a,b e Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b(mod 5).7. Suppose x,y e R. Then (x + y)² = x² + y² if and only if x = 0 or y = 0.9. Suppose a e 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,be Z. Prove that (a - 3)62 is even if and only if a is odd or b is even.13. Suppose a,b e Z. If a + b is odd, then a² +b² is odd.14. Suppose a e Z. Then a² | a if and only if a e{-1,0,1}.12, There exists a positive real number x for which x2 < Vã.15. Suppose a,b e Z. Prove that a +b is even if and only if a and b have the sameparity.16. Suppose a,b € 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 NSX.19. If n en 20 + 2' + 22 +23 +24 + ... +2&#34; = 2&#34;+1_1.h 11 (2&#34; – 1).in neNf= 0 is irrational.20. Theution o||n2 or21. Evfa|b and a | (b2 - c), then a c.22. Ifnd c are23. S

Accepted Answer

Answered: Image /qna-images/answer/548168f7-2c4d-441a-bedc-958dff28484a.jpg

8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b