556 CHAPTER 9 Mathematical Systems Exercises Warm Up Exercises In Exercises 1–10, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. In the mathematical system of clock 12 arithmetic, under the operation of addition, the set of elements is _______. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 { } 2. Since clock 12 arithmetic, under the operation of addition, contains only the elements 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 , { } the mathematical system is _______. Closed 3. In clock 12 arithmetic, since a a a 12 12 , + = + = for any element a, we say that 12 is the additive _______ element. Identity 4. In clock 12 arithmetic, since 1 11 11 1 12, + = + = we say that 1 and 11 are additive _______. Inverses 5. In clock 12 arithmetic, for any elements a b, , and c, a b c a b c ( ) ( ); + + = + + therefore, the _______ property holds under the operation of addition. Associative 6. In clock 12 arithmetic, for any elements a and b, a b b a; + = + therefore, the _______ property holds under the operation of addition. Commutative 7. In clock 12 arithmetic under the operation of addition, since the system is closed, there is an identity element, each element has an inverse, and the associative and commutative properties hold, the mathematical system is a(n) _______ group. Commutative 8. In a finite mathematical system, if every element does not appear in every row and every column of the table, you need to check the _______ property carefully. Associative 9. In a finite mathematical system defined by a table, if the elements are symmetric about the main diagonal, then the system is _______. Commutative 10. Groups that are not commutative are called noncommutative or _______ groups. Nonabelian Practice the Skills In Exercises 11–18, use Table 9.1 to determine the sum in clock 12 arithmetic. 11. 7 6 + 1 12. 9 8 + 5 13. 8 7 + 3 14. 11 7 + 6 15. 4 12 + 4 16. 12 12 + 12 17. 3 (8 9) + + 8 18. (8 7) 6 + + 9 In Exercises 19–26, determine the difference in clock 12 arithmetic by starting at the first number and counting counterclockwise on the clock the number of units given by the second number. 19. 10 3 − 7 20. 11 6 − 5 21. 4 10 − 6 22. 1 12 − 1 23. 6 10 − 8 24. 5 8 − 9 25. 5 5 − 12 26. 12 12 − 12 27. Use the following figure to develop an addition table for clock 6 arithmetic. The figure will also be used in Exercises 28–34. * 6 3 5 1 2 4 In Exercises 28–34, determine the sum or difference in clock 6 arithmetic. See Exercise 27. 28. 4 3 + 1 29. 6 2 + 2 30. 4 6 + 4 31. 5 2 − 3 32. 4 5 − 5 33. 3 (5 2) − − 6 34. 2 (4 5) − − 3 35. Use the following figure to develop an addition table for clock 7 arithmetic. The figure will also be used in Exercises 36– 43. * 7 6 5 4 3 2 1 In Exercises 36– 42, determine the sum or difference in clock 7 arithmetic. See Exercise 35. 36. 3 6 + 2 37. 4 4 + 1 38. 6 7 + 6 39. 5 5 + 3 40. 2 3 − 6 41. (4 5) 6 − − 7 42. 3 (2 6) − − 7 43. Determine whether clock 7 arithmetic under the operation of addition is a commutative group. Assume the associative property holds. Explain. Yes; it satisfies the five required properties. 44. Determine whether clock 6 arithmetic under the operation of addition is a commutative group. Assume the associative property holds. Explain. Yes; it satisfies the five required properties. *See Instructor Answer Appendix SECTION 9.2
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