CHAPTER 9 Test 583 Test CHAPTER 9 1. What is a mathematical system? A set of elements and a binary operation 2. List the requirements needed for a mathematical system to be a commutative group. Closure, identity element, inverses, associative property, commutative property 3. Is the set of integers a commutative group under the operation of addition? Explain your answer. Yes 4. Is the set of natural numbers a commutative group under the operation of subtraction? Explain your answer. No; not closed, not associative, not commutative 5. Develop a clock 5 arithmetic addition table. * 6. Is clock 5 arithmetic under the operation of addition a commutative group? Assume that the associative property holds. Explain your answer. Yes, it is a commutative group. 7. Consider the mathematical system u W S T R W T R W S S R W S T T W S T R R S T R W a) What is the binary operation? u b) Is this system closed? Explain. Yes c) Is there an identity element for this system under the given operation? Explain. Yes, T d) What is the inverse of the element R? S e) What is T R W ( ) ? u u S In Exercises 8 and 9, determine whether the mathematical system is a commutative group. Explain your answer completely. 8. 9. * a b c a a b c b b b a c c a d No, not closed, not associative ? 1 2 3 1 3 1 2 2 1 2 3 3 2 3 1 Yes, it is a commutative group. 10. Determine whether the mathematical system is a commutative group. Assume that the associative property holds. Explain your answer. ○ @ $ & % @ % @ $ & $ @ $ & % & $ & % @ % & % @ $ Yes, it is a commutative group. In Exercises 11 and 12, determine the modulo class to which the number belongs for the indicated modulo system. 11. 59, mod 4 3 12. 96, mod 11 8 In Exercises 13–16, determine all positive number replacements for the question mark, less than the modulus, that make the statement true. 13. ? 5 3 (mod 6) + ≡ 4 14. ? 3 4 (mod 5) − ≡ 2 15. 3 ? 7 (mod 8) − ≡ 4 16. 3 ? 2 (mod 6) ⋅ ≡ { } 17. a) Construct a modulo 5 multiplication table. * b) Is this mathematical system a commutative group? Explain your answer completely. No; no inverse for 0 In Exercises 18–20, given A 2 1 3 6 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and B 2 2 5 3 , = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ determine the following. 18. A B + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0 3 8 3 19. A B 2 3 − − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 10 4 9 21 20. A B × − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 7 36 12 *See Instructor Answer Appendix
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