582 CHAPTER 9 Mathematical Systems 16. In Exercises 5–8, determine the sum or difference in clock 12 arithmetic. 5. 9 8 + 5 6. 10 8 + 6 7. 8 10 − 10 8. 7 4 6 − + 9 9. List the properties of a group and explain what each property means. Closure, identity element, inverses, and associative property In Exercises 10–13, explain your answer. 10. Determine whether the set of integers under the operation of addition forms a group. Yes 11. Determine whether the set of integers under the operation of multiplication forms a group. No; no inverse for any integer except 1 and −1 12. Determine whether the set of natural numbers under the operation of addition forms a group. No; no identity element 13. Determine whether the set of rational numbers under the operation of multiplication forms a group. No; no inverse for 0 In Exercises 14–16, for the given mathematical system, determine which of the five properties of a commutative group do not hold. 15. ? 4 # L P 4 P 4 # L # 4 # L P L # L P 4 P L P 4 L Not every element has an inverse; not associative. For example, ≠ P P P P ( ? )?4 ?( ?4). 6 ρ σ ω ρ ρ ω σ σ ω σ ρ ω σ ρ ω No identity element; no inverses 14. ! ? p ! ? ! p ? p ? ! ! ? p p p ! p Not associative. For example, p p (! )?!( ?) ≠ 17. Consider the following mathematical system. Assume the associative property holds for the given operation. a) What are the elements of the set in this mathematical system? { , , , } b) What is the binary operation? c) Is the system closed? Explain. Yes d) Is there an identity element in the set? Yes; e) Does every element in the set have an inverse? If so, give each element and its corresponding inverse. Yes; – , – , – , – f) Give an example to illustrate the associative property. ( ) = ( ) g) Is the system commutative? Give an example. Yes; = h) Is this mathematical system a commutative group? Explain. Yes 9.3 In Exercises 18–23, determine the modulo class to which the number belongs for the indicated modulo system. 18. 19, mod 7 5 19. 24, mod 6 0 20. 47, mod 8 7 21. 34, mod 5 4 22. 71, mod 3 2 23. 54, mod 4 2 In Exercises 24–31, determine all positive number replacements (less than the modulus) for the question mark that make the statement true. 24. 2 5 ? (mod 6) + ≡ 1 25. 2 ? 3 (mod 7) − ≡ 6 26. ? 4 3 (mod 5) + ≡ 4 27. ? 3 2 (mod 4) − ≡ 1 28. 7 ? 5 (mod 8) ⋅ ≡ 3 29. ? 4 0 (mod 8) ⋅ ≡ 0, 2, 4, 6 30. ? 3 5 (mod 6) ⋅ = No solution 31. 4 ? 5 (mod7) ⋅ ≡ 3 32. Construct a modulo 6 addition table. Then determine whether the modulo 6 system forms a commutative group under the operation of addition. * 33. Construct a modulo 4 multiplication table. Then determine whether the modulo 4 system forms a commutative group under the operation of multiplication. * 34. Paramedic Shifts Julie, a paramedic, has the following work pattern. She works 3 days, has 4 days off, then works 4 days, then has 3 days off, and then the pattern repeats. Assume that today is the first day of the work pattern. a) Will Julie be working 30 days from today? Yes b) Julie’s niece is getting married 45 days from today. Will Julie have the day off? Yes 9.4 In Exercises 35–39, given A 2 1 3 0 = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and B 3 5 1 2 , = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ determine the following. 35. A B + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 5 4 2 2 36. A B − − − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 6 4 2 37. A B 3 2 − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 0 13 11 4 38. A B × − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 5 8 9 15 39. B A × − − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 9 3 4 1 *See Instructor Answer Appendix
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