554 CHAPTER 9 Mathematical Systems Mathematical Systems Without Numbers Thus far, all the systems we have discussed have been based on sets of numbers. Example 3 illustrates a mathematical system that uses symbols rather than numbers. Example 3 Investigating a System Without Numbers Use the mathematical system defined by Table 9.6 to determine a) the set of elements. b) the binary operation. c) whether the system is closed. d) the identity element. e) the inverse of ℮. f) (℮ e L) e ℮ and ℮ e (L e ℮). g) eL and eL Solution a) The set of elements of this mathematical system is {℮, L, }. b) The binary operation is e. c) Because the table does not contain any symbols other than ℮, and L, the system is closed under the binary operation, e. d) The identity element is . Locate the in the far-left column of Table 9.6. Note that the row of elements to the right of the matches the top row of elements. Also, locate the in the top row of elements. Note that the column of elements below the matches the far-left column of elements. We see that ℮ e = e ℮ = ℮, L e = e , = L L and e = e) To determine the inverse of ℮, locate ℮ in the far left column of Table 9.6. Now scan to the right of ℮ until you find the identity element . At the top of this column we see the element L. Since ℮ e e L L = ℮ L , = is the inverse of ℮. f) We first evaluate the expressions within the parentheses. (℮ e L) e ℮ e = ℮ = ℮ and ℮ e (L e ℮) = ℮ e = ℮ g) eL L = and eL L = 7 Now try Exercise 55 Table 9.6 A System Without Numbers e ℮ L ℮ L ℮ ℮ L L L ℮ Table 9.7 α β γ δ α δ α β γ β α β γ δ γ β γ δ α δ γ δ α β Table 9.8 Inverses Under Element Inverse = Identity Element α γ = β β β = β γ α = β δ δ = β Example 4 Is the System a Commutative Group? Determine whether the mathematical system defined by Table 9.7 is a commutative group under the operation of . Assume that the associative property holds for the given operation. Solution 1. Closure: The system is closed. 2. Identity element: The identity element is .β 3. Inverse elements: Each element has an inverse element, as illustrated in Table 9.8. 4. Associative property: It is given that the associative property holds. An example illustrating the associative property is ( ) ( ) ? ? α α α α γ δ = γ δ = γ γ δ = δ True 5. Commutative property: By examining Table 9.7, we can see that the elements are symmetric about the main diagonal. Thus, the system is commutative under the given operation, . One example of the commutative property is ? α α δ = δ γ = γ True All five properties are satisfied. Thus, the system is a commutative group. 7 Now try Exercise 57 Instructor Resources for Section 9.2 in MyLab Math • Objective-Level Videos 9.2 • PowerPoint Lecture Slides 9.2 • MyLab Exercises and Assignments 9.2
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