Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(10.) Suppose x,y, 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.

(10.) Suppose x,y, 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.

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
See similar textbooks
Question
Number 11
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. S. 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-420, then x 0. 9. Suppose n E Z. If 3{n2, then 3{n. 10. Suppose x,y,z € Z and x#0. If x{yz, then xy 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.beZz and a and b have the same parity, then 3a +7 and 76-4 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 8|1 (n2 – 1). 18) If a,be Z, then (a+ b) =a³ +b³ (mod 3). 19. Let a,b,CEZ and n eN. 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 =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 neN. If a =b (mod n), then a2 = ab (mod n). 24, If a b (mod n) and c = d (mod n), then ac = bd (mod n). 25. Let ne N. 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 15 od. 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 ged(a,n)=gcd(b,n). %3D godla 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. S. 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-420, then x 0. 9. Suppose n E Z. If 3{n2, then 3{n. 10. Suppose x,y,z € Z and x#0. If x{yz, then xy 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.beZz and a and b have the same parity, then 3a +7 and 76-4 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 8|1 (n2 – 1). 18) If a,be Z, then (a+ b) =a³ +b³ (mod 3). 19. Let a,b,CEZ and n eN. 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 =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 neN. If a =b (mod n), then a2 = ab (mod n). 24, If a b (mod n) and c = d (mod n), then ac = bd (mod n). 25. Let ne N. 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 15 od. 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 ged(a,n)=gcd(b,n). %3D godla b)
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS