A-31 ANSWERS A-31 Chapter 9 SECTION 9.1, PAGE 548 1. Binary 3. Closed 5. Identity 7. Inverse 9. Commutative 11. + = + a b b a for any elements a and + = + b; 3 4 4 3 35. 13. ⋅ ⋅ = ⋅ ⋅ a b c a b c ( ) ( ) for any elements a b , , and c; ⋅ ⋅ = ⋅ ⋅ (23)4 2(34) 15. 7 3 3 7 4 4 ? − = − ≠ − 17. (84)2 8(42) 1 4 ? ÷ ÷ = ÷ ÷ ≠ 19. a) Yes; the sum of any two integers is an integer. b) Yes; 0. c) Yes. d) + + = + + (12)3 (12)3 e) + = + 2 4 4 2 f) Yes; it satisfies the five properties needed. 21. a) Yes; the sum of any two positive integers is a positive integer. b) No. c) Since there is no identity element, each element does not have an inverse. d) + + = + + (23)4 2(34) e) + = + 1 2 2 1 f ) No; there is no identity element. 23. a) Yes; the product of any two positive integers is a positive integer. b) Yes; 1 c) No; other than 1, none of the elements have inverse elements because the set of positive integers does not contain fractions. d) (12)3 1(23) × × = × × e) × = × 4 3 3 4 f) No; not all elements have inverses. 25. Yes; it satisfies the four properties needed. 27. No; the system is not closed. 29. Yes; it satisfies the five properties needed. 31. No; there is no identity element. 33. No; the system is not closed. For example, 1 0 is undefined. 35. No; the system is not closed. For example, + − = 2 ( 2) 0, which is rational. There is also no identity element. 37. Answers will vary. SECTION 9.2, PAGE 556 1. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} 3. Identity 5. Associative 7. Commutative 9. Commutative 11. 1 13. 3 15. 4 17. 8 19. 7 21. 6 23. 8 25. 12 27. + 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 29. 2 31. 3 33. 6 + 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 37. 1 39. 3 41. 7 43. Yes; it satisfies the five required properties. 45. No 47. Yes; C is the identity element, since the row next to C is identical to the top row, and the column under C is identical to the far-left column. 49. a) The inverse of A is A, since A = A C. b) The inverse of B is B, since B = B C. c) The inverse of C is C, since C = C C. 51. No; the elements are not symmetric about the main diagonal. 53. a) {0, 2, 4, 6} b) Q c) Yes d) Yes; 0 e) Yes; 0–0, 2–6, 4–4, 6–2 f ) = Q Q Q Q (2 4) 62 (4 6) g) Yes; = Q Q 2 66 2 h) Yes 55. a) L {4, 5, } b) $ c) Yes d) Yes, L e) Yes, 4–5, 5–4, L L– f) (4 $ 5) $ 5 4 $ (5 $ 5) = g) Yes, = L L $ 4 4 $ h) Yes 57. a) Yes b) Yes; O c) G D O O L – , – , does not have an inverse, D G– d) L L D L L L D D ( ) ; ( ) q q q q = = e) No f) Yes g) No; not every element has an inverse and the associative property does not hold. 59. Not closed, not associative 61. No inverse for or for C, not associative 63. No identity element, no inverses, not associative, not commutative 65. a) + E O E E O O O E b) Yes, it is a commutative group; it satisfies the five properties. 67. a) It is closed; identity element is 6; inverses: 1–5, 2–2, 3–3, 4–4, 5–1, 6–6; is associative; for example, (2?5)?3 2?(5?3) 3? 3 2? 2 6 6 ? ? = = = b) 3?1 1? 3 2 4 ?= ≠
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