9.2 Finite Mathematical Systems 553 Example 2 A Finite Mathematical System Consider the mathematical system defined by Table 9.4. Assume that the associative property holds for the given operation. a) List the elements in the set of this mathematical system. b) Identify the binary operation. c) Determine whether this mathematical system is a commutative group. Solution a) The set of elements for this mathematical system consists of the elements found in both the top row and the far-left column of the table: 2, 4, 6, 8 . { } b) The binary operation is © . c) We will determine whether the five requirements for a commutative group are satisfied. 1. Closure : All the elements in the table are in the original set of elements 2, 4, 6, 8 , { } so the mathematical system is closed. 2. Identity element : The identity is 6. Note that the row of elements to the right of the 6 is identical to the top row and the column of elements under the 6 is identical to the far-left column. 3. Inverse elements : When an element operates on its inverse element, the result is the identity element. For this example, the identity is 6. To determine the inverse of 2, determine which element from the set, 2, 4, 6, or 8 { } replaces the shaded area in the following equation and gives a true statement: 2 6 © = . From Table 9.4, we see that 2 2 6. © = Thus, 2 is its own inverse. To determine the inverse of 4, determine which element replaces the shaded area in the following equation and gives a true statement: 4 6 © = . From Table 9.4, we see that 4 8 8 4 6. © = © = Therefore, 4 is the inverse of 8, and 8 is the inverse of 4. The elements and their inverses are shown in Table 9.5. Note that every element in the mathematical system has an inverse. Now try Exercise 53 Table 9.4 A Four-Element System © 2 4 6 8 2 6 8 2 4 4 8 2 4 6 6 2 4 6 8 8 4 6 8 2 Table 9.5 Inverses Under © Element © Inverse = Identity Element 2 © 2 = 6 4 © 8 = 6 6 © 6 = 6 8 © 4 = 6 4. Associative property : It is given that the associative property holds for this binary operation, © . One example of the associative property is (2 8) 4 2 (8 4) 4 4 2 6 2 2 ? ? © © = © © © = © = True 5. Commutative property : The elements in Table 9.4 are symmetric about the main diagonal, so the commutative property holds for the operation of © . One example of the commutative property is 2 4 4 2 8 8 ? © = © = True The five necessary properties hold. Thus, the mathematical system is a commutative group. 7 Timely Tip When determining whether a finite mathematical system is a group, remember the following tips. Closure: The top row and the far-left column of the table will show the elements in the set. If any other elements appear in the body of the table, then the system is not closed. Identity: Look for a row in the table that matches the top row and a column that matches the far-left column. The element that corresponds to this row and this column is the identity element. Inverses: To determine the inverse of an element, a, first find a in the far-left column of the table. Next, scan across the row that corresponds to a until you find the identity element. The number at the top of this column is the inverse of a. Associative Property: If there is a row or column of the table that does not contain every element in the system, you must check the associative property carefully. Commutative Property: If the elements are symmetric about the main diagonal, then the system is commutative.
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