9.1 Groups 549 Problem Solving In Exercises 25–34, explain your answer. 25. Is the set of rational numbers a group under the operation of addition? Yes; it satisfies the four properties needed. 26. Is the set of positive rational numbers a commutative group under the operation of multiplication? Yes; it satisfies the five properties needed. 27. Is the set of negative integers a group under the operation of multiplication? No; the system is not closed. 28. Is the set of negative integers a group under the operation of division? No; the system is not closed. 29. Is the set of positive real numbers a commutative group under the operation of multiplication? Yes; it satisfies the five properties needed. 30. Is the set of whole numbers a commutative group under the operation of multiplication? No; not all elements have inverses. There is no inverse for any whole number except for 1. 31. Is the set of negative integers a commutative group under the operation of addition? No; there is no identity element. 32. Is the set of integers a group under the operation of multiplication? No; not all elements have inverses. There is no inverse for any integers except 1 and 1. − 33. Is the set of rational numbers a commutative group under the operation of division? No; the system is not closed. For example, 1 0 is undefined. 34. Is the set of rational numbers a group under the operation of subtraction? No; it does not have an identity element and does not satisfy the associative property. Challenge Problems/Group Activities In Exercises 35 and 36, explain your answer. 35. Is the set of irrational numbers a group under the operation of addition? No; the system is not closed. For example, 2 ( 2) 0, + − = which is rational. There is also no identity element. 36. Is the set of irrational numbers a group under the operation of multiplication? No; the system is not closed. For example, 2 2 2, ⋅ = which is rational. 37. Create a mathematical system with two binary operations. Select a set of elements and two binary operations so that one binary operation with the set of elements meets the requirements for a group and the other binary operation does not. Explain why the one binary operation with the set of elements is a group. For the other binary operation and the set of elements, provide counterexamples to show that it is not a group. Answers will vary. Research Activity 38. Rings and Fields There are other classifications of mathematical systems besides groups. For example, there are rings and fields. Determine the requirements that must be met for a mathematical system to be (a) a ring and (b) a field. (c) Is the set of real numbers, under the operations of addition and multiplication, a field? *See Instructor Answer Appendix d) The associative property holds. Provide one example to demonstrate the associative property. (23)4 2(34) + + = + + e) The commutative property holds. Provide one example to demonstrate the commutative property. 1 2 2 1 + = + f) Is this mathematical system a commutative group? Explain. No; there is no identity element. 22. Consider the set of whole numbers under the operation of addition. a) Is the system closed? Explain. Yes; the sum of any two whole numbers is a whole number. b) Is there an identity element? If so, what is it? Yes; 0. c) Does each element in the set have an inverse? Explain. No; other than 0, none of the elements have inverse elements because the set of whole numbers does not contain negative integers. d) The associative property holds. Provide one example to demonstrate the associative property. (45)6 4(56) + + = + + e) The commutative property holds. Provide one example to demonstrate the commutative property. 4 5 5 4 + = + f) Is this mathematical system a commutative group? Explain. No; not all elements have inverses. 23. Consider the set of positive integers under the operation of multiplication. a) Is the system closed? Explain. Yes; the product of any two positive integers is a positive integer. b) Is there an identity element? If so, what is it? Yes; 1. c) Does each element in the set have an inverse? Explain. No; other than 1, none of the elements have inverse elements because the set of positive integers does not contain fractions. d) The associative property holds. Provide one example to demonstrate the associative property. ⋅ ⋅ = ⋅ ⋅ (12)3 1(23) e) The commutative property holds. Provide one example to demonstrate the commutative property. ⋅ = ⋅ 4 3 3 4 f) Is this mathematical system a commutative group? No; not all elements have inverse elements. 24. Consider the set of real numbers under the operation of multiplication. a) Is the system closed? Explain. Yes; the product of any two real numbers is a real number. b) Is there an identity element? If so, what is it? Yes; 1. c) Does each element in the set have an inverse? Explain. No; not all elements have inverse elements. 0 does not have an inverse element because 0 does not have a reciprocal. d) The associative property holds. Provide one example to demonstrate the associative property. (23)4 2(34) ⋅ ⋅ = ⋅ ⋅ e) The commutative property holds. Provide one example to demonstrate the commutative property. 2 3 3 2 ⋅ = ⋅ f) Is this mathematical system a commutative group? No; not all elements have inverse elements.
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