548 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. A mathematical system consists of a set of elements and at least one _______ operation. Binary 2. A binary operation is an operation or rule that can be performed on exactly two elements of a set, with the result being a(n) _______ element. Single 3. If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is _______ under the given binary operation. Closed 4. A specific example illustrating that a property is not true is called a(n) _______. Counterexample 5. Since the sum of 0 and any integer is the given integer, we say that 0 is the additive _______ element for the set of the integers under the operation of addition. Identity 6. Since the product of 1 and any integer is the given integer, we say that 1 is the _______ identity element for the set of the integers under the operation of multiplication. Multiplicative 7. When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the _______ of the other. Inverse 8. If a mathematical system possesses the following properties—closure, identity element, inverses, and the associative property—then the mathematical system is a(n) _______. Group 9. A group that also satisfies the commutative property is called a(n) _______ (or abelian) group. Commutative 10. Another name for a commutative group is a(n) _______ group. Abelian Practice the Skills 11. Give the commutative property of addition and illustrate the property with an example. a b b a + = + for any elements a and b; 3 4 4 3 + = + 12. Give the commutative property of multiplication and illustrate the property with an example. a b b a ⋅ = ⋅ for any elements a and b; 2 3 3 2 ⋅ = ⋅ 13. Give the associative property of multiplication and illustrate the property with an example. a b c a b c ( ) ( ) ⋅ ⋅ = ⋅ ⋅ for any elements a b , , and c;(23)4 2(34) ⋅ ⋅ = ⋅ ⋅ 14. Give the associative property of addition and illustrate the property with an example. a b c a b c ( ) ( ) + + = + + for any elements a b , , and c;(34)5 3(45) + + = + + 15. Give an example to show that the commutative property does not hold for the set of integers under the operation of subtraction. 7 3 3 7 4 4 ? − = − ≠ − 16. Give an example to show that the commutative property does not hold for the set of integers under the operation of division. ÷ = ÷ ≠ 4 2 2 4 2 1 2 ? 17. Give an example to show that the associative property does not hold for the set of integers under the operation of division. ÷ ÷ = ÷ ÷ ≠ (84)2 8(42) 1 4 ? 18. Give an example to show that the associative property does not hold for the set of integers under the operation of subtraction. * 19. Consider the set of integers under the operation of addition. a) Is the system closed? Explain. Yes; the sum of any two integers is an integer. b) Is there an identity element? If so, what is it? Yes; 0. c) Does each element in the set have an inverse? Explain. Yes; the opposite of each element is its inverse. d) The associative property holds. Provide one example to demonstrate the associative property. (12)3 (12)3 + + = + + e) The commutative property holds. Provide one example to demonstrate the commutative property. 2 4 4 2 + = + f) Is this mathematical system a commutative group? Explain. Yes; it satisfies the five properties needed. 20. Consider the set of real numbers under the operation of addition. a) Is the system closed? Explain. Yes; the sum of any two real numbers is a real 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. Yes; the opposite of each element is its inverse. d) The associative property holds. Provide one example to demonstrate the associative property. π π + + = + + ( 2) 3 (2 3) e) The commutative property holds. Provide one example to demonstrate the commutative property. π π + = + 1 1 f) Is this mathematical system a commutative group? Explain. Yes; it satisfies the five properties needed. 21. Consider the set of positive integers under the operation of addition. a) Is the system closed? Explain. Yes; the sum of any two positive integers is a positive integer. b) Is there an identity element? If so, what is it? No. c) Does each element in the set have an inverse? Explain. Since there is no identity element, each element does not have an inverse. SECTION 9.1 *See Instructor Answer Appendix
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