558 CHAPTER 9 Mathematical Systems e) Does the associative property hold for the system? No f) Does the commutative property hold for the system? Yes g) Is the mathematical system a commutative group? Explain. No; not every element has an inverse and the associative property does not hold. 58. Consider the mathematical system defined by the following table. ✦ P S T L P S L P S S L P S T T P S T L L S T L P a) Is the system closed? Explain. Yes b) Is there an identity element in the set? If so, what is it? Yes; T c) For each element in the set, give the corresponding inverse element, if it exists. P does not have an inverse, S L T T L S –, –, – d) Evaluate (P ✦ L) ✦ L and P ✦ (L ✦ L). (P ✦ L )✦ L T; P ✦ (L ✦ L) S e) Does the associative property hold for the system? No f) Does the commutative property hold for the system? Yes g) Is the mathematical system a commutative group? Explain. No; not every element has an inverse and the associative property does not hold. In Exercises 59–64, for the mathematical system given, determine which of the five properties of a commutative group do not hold. 59. β Δ Φ Γ Δ Δ Φ Γ Φ Φ Γ Δ Γ Γ Δ Π Not closed, not associative: β β β β Φ Φ Γ ≠ Φ Φ Γ ( ) ( ) 60. ⊗ M M M M M * 61. C ? T P T P C C C P C T ? C ? T P T C T P ? P C T P ? No inverse for or for C, not associative C C (P ) P ( ) ≠ 62. *See Instructor Answer Appendix a b π 0 Δ a Δ a b π 0 b a b π 0 Δ π b π 0 Δ a 0 π 0 Δ a b Δ 0 Δ π b a Not associative, a( a) a Δ ≠ a( ); Δ not commutative, Δ π π ≠ Δ 63. F a b c d e a c d e a b b d e a b c c e a b c d d a b c e d e b c d e a No identity element, no inverses, not associative, not commutative 64. 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 1 2 3 4 5 2 0 2 4 0 2 4 3 0 3 0 3 0 3 4 0 4 2 0 4 2 5 0 5 4 3 2 1 0, 2, 3, and 4 do not have an inverse. Problem Solving 65. a) Consider the set consisting of two elements E O, , { } where E stands for an even number and O stands for an odd number. For the operation of addition, complete the table. * + E O E O b) Determine whether this mathematical system forms a commutative group under the operation of addition. Explain your answer. Yes, it is a commutative group; it satisfies the five properties. 66. a) Let E and O represent even numbers and odd numbers, respectively, as in Exercise 65. Complete the table for the operation of multiplication. * × E O E O b) Determine whether this mathematical system forms a commutative group under the operation of multiplication. Explain your answer. Not a group, E has no inverse.
RkJQdWJsaXNoZXIy NjM5ODQ=