9.2 Finite Mathematical Systems 557 In Exercises 45 and 46, determine if the system is closed. Explain how you determined your answer. 45. c) Is the system closed? Explain. Yes d) Is there an identity element in the set? If so, what is it? Yes; 0 e) Does every element in the set have an inverse? If so, give each element and its corresponding inverse. Yes; 0–0, 2–6, 4–4, 6–2 f) Give an example to illustrate the associative property. Q Q Q Q (2 4) 6 2 (4 6) = g) Is the system commutative? Give an example to verify your answer. Yes; Q Q 2 6 6 2 = h) Is the mathematical system a commutative group? Explain. Yes In Exercises 54–56, repeat parts (a) – (h) of Exercise 53 for the mathematical system defined by the given table. Assume that the associative property holds for the given operation. 54. ⊗ x y z x x y z y y z z z z t y No 46. a b c a a b c b b c a c c a b Yes In Exercises 47 and 48, determine if the system has an identity element. If so, state the identity element. Explain how you determined your answer. 47. W A B C A C B A B B C B C A B C * 48. h h h h h There is no identity element. In Exercises 49 and 50, the identity element is C. Determine the inverse, if it exists, of (a) A, (b) B, and (c) C. 49. A B C A C B A B B C B C A B C * 50. ⊗ A B C A A A A B A C B C A B C * In Exercises 51 and 52, determine if the system is commutative. Explain how you determined your answer. 51. I P A L P L P A A P L A L A L P No; the elements are not symmetric about the main diagonal. 52. A Z W A Z W A Z W A Z W A Z Z Yes; the elements are symmetric about the main diagonal. 53. Consider the mathematical system defined by the following table. Assume that the associative property holds for the given operation. Q 0 2 4 6 0 0 2 4 6 2 2 4 6 0 4 4 6 0 2 6 6 0 2 4 a) What are the elements of the set in this mathematical system? 0, 2, 4, 6 { } b) What is the binary operation? Q ® ♣ ♦ ♥ ♠ ♣ ♥ ♠ ♣ ♦ ♦ ♠ ♣ ♦ ♥ ♥ ♣ ♦ ♥ ♠ ♠ ♦ ♥ ♠ ♣ a) ♣, ♦, ♥, ♠} b) ® c) Yes d) Yes; ♥ e) Yes; ♣–♣, ♦–♠, ♥–♥, ♠–♦ f) (♣ ® ♦) ® ♥ = ♣ ® (♦ ® ♥) g) Yes; ♣ ® ♦ = ♦ ® ♣ h) Yes 55. $ 4 5 L 4 5 L 4 5 L 4 5 L 4 5 L * 56. t 3 5 8 4 3 5 8 4 3 5 8 4 3 5 8 4 3 5 8 4 3 5 8 4 * 57. Consider the mathematical system defined by the following table. ✰ G O L D G L G D O O G O L D L D L D G D O D G L a) Is the system closed? Explain. Yes b) Is there an identity element in the set? If so, what is it? Yes; O c) For each element in the set, give the corresponding inverse element, if it exists. G D O O L – , – , does not have an inverse, D G– d) Evaluate (L ✰ L) ✰ D and L ✰ (L ✰ D). (L ✰ L) ✰ D L; L ✰ (L ✰ D) D *See Instructor Answer Appendix
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