9.2 Finite Mathematical Systems 551 The binary operation of this mathematical system, ,+ is defined by Table 9.1. To determine the value of a b, + where a and b are any two numbers in the set, find a in the left-hand column and find b along the top row. Assume that there is a horizontal line through a and a vertical line through b; the point of intersection of these two lines is where you find the value of a b. + For example, 10 4 2 + = has been circled in Table 9.1. Note that 4 10 + also equals 2, but this result will not necessarily hold for all examples in this chapter. Did You Know? Twenty-Four-Hour Clock Systems The earliest mechanical clocks showed 24 hour markings on the clock. Many surviving examples of 24-hour clocks still exist around the world. An example is shown above. Most non-Englishspeaking countries use a 24-hour clock system—some in conjunction with a 12-hour clock system. In the 24-hour clock system, midnight is 00:00, 1 A.M. is 1:00, 2 A.M. is 2:00, and noon is 12:00. And, 1 P.M. is 13:00, 2 P.M. is 14:00, and so on. In the United States and Canada, the 24-hour clock system is commonly referred to as “military time” for its use in the military. The 24-hour clock system is also used in hospitals to avoid confusion between times for medicine or treatment. Learning Catalytics Keyword: Angel-SOM-9.2 (See Preface for additional details.) Example 1 A Commutative Group? Determine whether the clock 12 arithmetic system under the operation of addition is a commutative group. Solution Check the five requirements that must be satisfied for a commutative group. 1. Closure : Is the set of elements in clock 12 arithmetic closed under the operation of addition? Yes it is, since Table 9.1 contains only the elements in the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 . { } If Table 9.1 had contained an element other than the numbers 1 through 12, the set would not have been closed under addition. 2. Identity element : Is there an identity element for clock 12 arithmetic? If the time is currently 4 o’clock, how many hours have to pass before it is 4 o’clock again? Twelve hours: 4 12 12 4 4. + = + = In fact, given any hour, in 12 hours the clock will return to the starting point. Therefore, 12 is the additive identity element in clock 12 arithmetic. In examining Table 9.1, we see that the row of numbers next to the 12 in the far-left column is identical to the top row of numbers. We also see that the column of numbers under the 12 in the top row is identical to the column of numbers on the far left. The search for such a column and row is one technique for determining whether an identity element exists for a system defined by a table. 3. Inverse elements : Is there an inverse for the number 4 in clock 12 arithmetic for the operation of addition? Recall that the identity element in clock 12 arithmetic is 12. What number when added to 4 gives 12, that is, 4 12? + = . Table 9.1 shows that 4 8 12 + = and also that 8 4 12. + = Thus, 8 is the additive inverse of 4, and 4 is the additive inverse of 8. To determine the additive inverse of 7, find 7 in the far-left column of Table 9.1. Look to the right of the 7 until you come to the identity element 12. Then determine the number at the top of this column. The number is 5. Since 7 5 5 7 12, 5 + = + = is the inverse of 7 and 7 is the inverse of 5. The other inverses can be determined in the same way. Table 9.2 shows each element in clock 12 arithmetic and its inverse. Note that each element in the set has an inverse. Table 9.2 Clock 12 Inverses Element + Inverse = Identity Element 1 + 11 = 12 2 + 10 = 12 3 + 9 = 12 4 + 8 = 12 5 + 7 = 12 6 + 6 = 12 7 + 5 = 12 8 + 4 = 12 9 + 3 = 12 10 + 2 = 12 11 + 1 = 12 12 + 12 = 12 Julia Lopatina/Shutterstock
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