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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 90E
Related questions
Question
Please a detailed prove for number 4,8 and 10 and why that approach was chosen
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning