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7. Suppose a,beZ. If both ab and a +b are even, then both a and b are even. 4. Suppose a,b,c € Z. If a does not divide be, then a does not divide b. 2. Suppose ne Z. If n2 is odd, then n is odd. 3. Suppose a,be Z. If a (b² -2b) is odd, then a and b are odd a 5. Suppose x ER. If x + 5x<0 then x<0. 6. Suppose x ER. If x-x> 0 then x>-1. 8. Suppose x €R. If x - 4x*+3x3 -x2 + 3x - 420, then x20. 9. Suppose n e Z. If 3{n2, then 3 n. 10, Supposex,y,z €Z and x#0. If x{yz, then xy andx z. 11. Suppose x,y€ Z. If x²(y+3) is even, then x is even or y is odd. 12. Suppose a € Z. If a? is not divisible by 4, then a is odd. 13. Suppose x€R. If x+7x³ + 5x >x* +x2 + 8, then x > 0. B. Prove the following statements using either direct or contrapositive proof 14. If a,beZ and a and b have the same parity, then 3a +7 and 76 -4 do not. 15. Suppose xe Z. If x – 1 is even, then x is odd. 16. Suppose x,y € Z. If x+ y is even, then x and y have the same parity. 17. If n is odd, then 8| (n2 – 1). 18 If a,beZ, then (a+ b)³ = a³ + b³ (mod 3). 19. Let a,b,CeZ and neN. If a = b (mod n) and a = c (mod n), then c =b (mod n). 20. If a eZ and a = 1 (mod 5), then a² = 1 (mod 5). 21. Let a,be Z and ne N. If a =b (mod n), then a = 63 (mod n). 22. Let a e Z, neN. If a has remainder r when divided by n, then a =r (mod n). 23. Let a,be Z and neN. If a =b (mod n), then a2 = ab (mod n).

7. Suppose a,beZ. If both ab and a +b are even, then both a and b are even. 4. Suppose a,b,c € Z. If a does not divide be, then a does not divide b. 2. Suppose ne Z. If n2 is odd, then n is odd. 3. Suppose a,be Z. If a (b² -2b) is odd, then a and b are odd a 5. Suppose x ER. If x + 5x<0 then x<0. 6. Suppose x ER. If x-x> 0 then x>-1. 8. Suppose x €R. If x - 4x*+3x3 -x2 + 3x - 420, then x20. 9. Suppose n e Z. If 3{n2, then 3 n. 10, Supposex,y,z €Z and x#0. If x{yz, then xy andx z. 11. Suppose x,y€ Z. If x²(y+3) is even, then x is even or y is odd. 12. Suppose a € Z. If a? is not divisible by 4, then a is odd. 13. Suppose x€R. If x+7x³ + 5x >x* +x2 + 8, then x > 0. B. Prove the following statements using either direct or contrapositive proof 14. If a,beZ and a and b have the same parity, then 3a +7 and 76 -4 do not. 15. Suppose xe Z. If x – 1 is even, then x is odd. 16. Suppose x,y € Z. If x+ y is even, then x and y have the same parity. 17. If n is odd, then 8| (n2 – 1). 18 If a,beZ, then (a+ b)³ = a³ + b³ (mod 3). 19. Let a,b,CeZ and neN. If a = b (mod n) and a = c (mod n), then c =b (mod n). 20. If a eZ and a = 1 (mod 5), then a² = 1 (mod 5). 21. Let a,be Z and ne N. If a =b (mod n), then a = 63 (mod n). 22. Let a e Z, neN. If a has remainder r when divided by n, then a =r (mod n). 23. Let a,be Z and neN. If a =b (mod n), then a2 = ab (mod n).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 5E: Suppose a codding scheme is devised that maps -bit words onto -bit code words. The efficiency of the...
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31. Suppose the division algorithm applied to a and b yields a = qb + r. Prove about how a direct proof would work. In most cases contrapositive is easier.) A. Prove the following statements with contrapositive proof. (In each case, think Contrapositive Proof 136 Exercises for Chapter 5 1. Suppose neZ. If n is even, then n is even. 2. Suppose n eZ. If n2 is odd, then n is odd. 3. Suppose a, be Z. If a2(b² -2b) is odd, then a and b are odd 4. Suppose a, b,c € Z. If a does not divide be, then a does not divid,. 5. Suppose x ER. If x2 + 5x<0 then x<0. 6. Suppose x ER. If x -x> 0 then x> -1. 7. Suppose a, beZ. If both ab and a +b are even, then both a and 8. Suppose x ER. If x - 4x* +3x -x² + 3x -4>0, then x >0. 9. Suppose n e Z. If 3{n², then 3łn. 10. Suppose x,y,z € Z and x#0. If x{yz, then x{y and x z. 11. Suppose x,y€ Z. If x²(y+3) is even, then x is even or y is odd. 12. Suppose a € Z. If a² is not divisible by 4, then a is odd. 13. Suppose x € R. If x³ + 7x³ + 5x >x* + x2 + 8, then x > 0. B. Prove the following statements using either direct or contrapositive proof. 14. If a.be Zz and a and b have the same parity, then 3a + 7 and 7b-4 do not. 15. Suppose x€ Z. If x³ –1 is even, then x is odd. 16. Suppose x, y € Z. If x+ y is even, then x and y have the same parity. 17. Ifn is odd, then 81 (n2 – 1). 18) If a,be Z, then (a + b)³ = a3 + b³ (mod 3). 19. Let a,b, CeZ and neN. If a = b (mod n) and a = c (mod n), then c =b (mod n). 20. If a eZ and a = 1 (mod 5), then a2 = 1 (mod 5). 21. Let a,beZ and neN. If a = b (mod n), then a = b3 (mod n). 22. Let a e Z, neN. If a has remainder r when divided by n, then a =r (mod n). 23. Let a,be Z and n eN. If a =b (mod n), then a? = ab (mod n). (24, If a =b (mod n) and c = d (mod n), then ac = bd (mod n). 25. Let neN. If 2"-1 is prime, then n is prime. 26. If n 2-1 for keN, then every entry in Row n of Pascal's Triangle 1s odr 27. If a = 0 (mod 4) or a = 1 (mod 4), then () is even. 28. If ne Z, then 4 (n2-3). 29. If integers a and b are not both zero, then ged(a, b) = gcd(a-0,01. 30. If a = b (mod n), then gcd(a,n)3 gcd(b,n). %3D gedla b)
Transcribed Image Text:31. Suppose the division algorithm applied to a and b yields a = qb + r. Prove about how a direct proof would work. In most cases contrapositive is easier.) A. Prove the following statements with contrapositive proof. (In each case, think Contrapositive Proof 136 Exercises for Chapter 5 1. Suppose neZ. If n is even, then n is even. 2. Suppose n eZ. If n2 is odd, then n is odd. 3. Suppose a, be Z. If a2(b² -2b) is odd, then a and b are odd 4. Suppose a, b,c € Z. If a does not divide be, then a does not divid,. 5. Suppose x ER. If x2 + 5x<0 then x<0. 6. Suppose x ER. If x -x> 0 then x> -1. 7. Suppose a, beZ. If both ab and a +b are even, then both a and 8. Suppose x ER. If x - 4x* +3x -x² + 3x -4>0, then x >0. 9. Suppose n e Z. If 3{n², then 3łn. 10. Suppose x,y,z € Z and x#0. If x{yz, then x{y and x z. 11. Suppose x,y€ Z. If x²(y+3) is even, then x is even or y is odd. 12. Suppose a € Z. If a² is not divisible by 4, then a is odd. 13. Suppose x € R. If x³ + 7x³ + 5x >x* + x2 + 8, then x > 0. B. Prove the following statements using either direct or contrapositive proof. 14. If a.be Zz and a and b have the same parity, then 3a + 7 and 7b-4 do not. 15. Suppose x€ Z. If x³ –1 is even, then x is odd. 16. Suppose x, y € Z. If x+ y is even, then x and y have the same parity. 17. Ifn is odd, then 81 (n2 – 1). 18) If a,be Z, then (a + b)³ = a3 + b³ (mod 3). 19. Let a,b, CeZ and neN. If a = b (mod n) and a = c (mod n), then c =b (mod n). 20. If a eZ and a = 1 (mod 5), then a2 = 1 (mod 5). 21. Let a,beZ and neN. If a = b (mod n), then a = b3 (mod n). 22. Let a e Z, neN. If a has remainder r when divided by n, then a =r (mod n). 23. Let a,be Z and n eN. If a =b (mod n), then a? = ab (mod n). (24, If a =b (mod n) and c = d (mod n), then ac = bd (mod n). 25. Let neN. If 2"-1 is prime, then n is prime. 26. If n 2-1 for keN, then every entry in Row n of Pascal's Triangle 1s odr 27. If a = 0 (mod 4) or a = 1 (mod 4), then () is even. 28. If ne Z, then 4 (n2-3). 29. If integers a and b are not both zero, then ged(a, b) = gcd(a-0,01. 30. If a = b (mod n), then gcd(a,n)3 gcd(b,n). %3D gedla b)
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