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

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

Advanced Engineering Mathematics
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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Number 8
Constructive Versus Non-Constructive Proofs This existence proof has inside of it a separate proof (by contradiction) in other books and articles, as well as to the possibility of crafting such Be alert to constructive and non-constructive proofs as you read proofs that log2 9 is irrational. Such combinations of proof techniques are, of course, tements 155 uctive. corem; ing it. typical. exist proofs of your own. iomsato onal, Exercises for Chapter 7 Drove the following statements. These exercises are cumulative, covering all ear we techniques 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 same parity. 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 e n 20 + 2' + 22 +23 +24 + ... +2" = 2"+1_1. h 11 (2" – 1). in neNf = 0 is irrational. 20. The ution o ||n2 or 21. Ev fa|b and a | (b2 - c), then a c. 22. If nd c are 23. S
Transcribed Image Text:Constructive Versus Non-Constructive Proofs This existence proof has inside of it a separate proof (by contradiction) in other books and articles, as well as to the possibility of crafting such Be alert to constructive and non-constructive proofs as you read proofs that log2 9 is irrational. Such combinations of proof techniques are, of course, tements 155 uctive. corem; ing it. typical. exist proofs of your own. iomsato onal, Exercises for Chapter 7 Drove the following statements. These exercises are cumulative, covering all ear we techniques 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 same parity. 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 e n 20 + 2' + 22 +23 +24 + ... +2" = 2"+1_1. h 11 (2" – 1). in neNf = 0 is irrational. 20. The ution o ||n2 or 21. Ev fa|b and a | (b2 - c), then a c. 22. If nd c are 23. S
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