ANSWERS A-33 34. a) Yes b) Yes 35. − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 5 4 2 2 36. − − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 6 4 2 37. − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0 13 11 4 38. − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 5 8 9 15 39. − − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 9 3 4 1 CHAPTER TEST, PAGE 583 1. A set of elements and a binary operation. 2. Closure, identity element, inverses, associative property, commutative property 3. Yes 4. No; not closed, not associative, not commutative, 5. 55. ⎡ ⎣ ⎤ ⎦ 36.04 47.52 57. a) Yes b) Yes, ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 0 0 0 0 0 0 c) Yes d) Answers will vary. e) Yes f) Yes 59. Answers will vary. 61. Answers will vary. 63. ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ a b c d a b c d a b c d 1 0 0 1 1 0 0 1 REVIEW EXERCISES, PAGE 581 1. A binary operation is an operation that can be performed on two and only two elements of a set. The result is a single element. 2. A mathematical system consists of a set of elements and at least one binary operation. 3. No; for example ÷ = 2 3 2 3 and 2 3 is not a whole number. 4. Yes; the difference of any two real numbers is a real number. 5. 5 6. 6 7. 10 8. 9 9. Closure, identity element, inverses, and associative property 10. Yes 11. No; no inverse for any integer except 1 and −1 12. No; no identity element 13. No; no inverse for 0 14. No identity element; no inverses 15. Not every element has an inverse; not associative. For example, P P P P ( ? )?4 ?( ?4). ≠ 16. Not associative. For example, ≠ u u u u p p (! ) ? ! ( ?) 17. a) { , , , } b) c) Yes d) Yes; e) Yes; – , – , – , – f) ( ) = ( ) g) Yes; = h) Yes 18. 5 19. 0 20. 7 21. 4 22. 2 23. 2 24. 1 25. 6 26. 3 27. 0, 2, 4, 6 28. 0 29. 9 30. No solution 31. 5 32. + 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4 Yes; it is a commutative group. 33. × 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 No; no inverse for 0 or 2 + 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6. Yes; it is a commutative group. 7. a) u b) Yes c) Yes, T d) S e) S 8. No, not closed, not associative. 9. Yes; it is a commutative group. 10. Yes; it is a commutative group. 11. 3 12. 8 13. 4 14. 2 15. 4 16. {} 17. a) × 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 b) No; no inverse for 0 18. − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0 3 8 3 19. − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 10 4 9 21 20. − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 7 36 12 Chapter 10 SECTION 10.1, PAGE 592 1. 100 3. 100 5. Previous 7. 40% 9. 35% 11. 0.8% 13. 378.0% 15. 0.09 17. 0.0724 19. 0.0025 21. 1.359 23. $11,250 25. 2.56 27. 25% 29. 8% 31. 300 33. 150 35. 25% 37. 4.7 grams 39. 471 comedy movies 41. 192 horror movies 43. 21.46 m of oxygen 45. 9.7 m of chlorine 47. 47.0% 49. 14.0% 51. 8.7% 53. 81.0% 55. a) $3.81 b) $9.53 c) $76.84 57. $50.25 59. 12 students 61. $42,265 63. 5.3% decrease 65. 40% 67. 30.3% 69. a) $68,750 b) $206,250 71. He will have a loss of $10. 73. a) $145.34 b) $142.49 c) b)
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