550 CHAPTER 9 Mathematical Systems Suppose you decide to use your slow cooker to cook a meal for a dinner party. You would like to eat at 6:30 P.M., and it will take 10 hours for the meal to cook. What time should you begin to cook the meal in the slow cooker? Or suppose you need to take medicine every eight hours. You last took the medicine at 7:45 A.M. When should you take the medicine again? Questions such as these involve the use of mathematical systems that we will study in this section. Finite Mathematical Systems SECTION 9.2 LEARNING GOALS Upon completion of this section, you will be able to: 7 Determine whether a finite mathematical system defined by clock arithmetic is a group. 7 Determine whether a finite mathematical system without numbers is a group. Why This Is Important Timekeeping is just one application of finite mathematical systems. In the preceding section, we presented infinite mathematical systems. In this section, we present some finite mathematical systems. A finite mathematical system is one whose set contains a finite number of elements. Clock Arithmetic Let’s develop a finite mathematical system called clock 12 arithmetic. The set of elements in this system will be the hours on a clock: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 . { } The binary operation that we will use is addition, which we define as movement of the hour hand in a clockwise direction. Assume that it is 4 o’clock. What time will it be in 9 hours? (See Fig. 9.1.) If we add 9 hours to 4 o’clock, the clock will read 1 o’clock. Thus, 4 9 1 + = in clock arithmetic. Would 9 4 + be the same as + 4 9? Yes it will because 4 9 9 4 1. + = + = Table 9.1 is the addition table for clock arithmetic. Its elements are based on the definition of addition as previously illustrated. For example, the sum of 4 and 9 is 1, so we put a 1 in the table where the row to the right of the 4 intersects the column below the 9 (see the pink shading). Likewise, the sum of 11 and 10 is 9, so we put a 9 in the table where the row to the right of the 11 intersects the column below the 10 (see the red outline). 1 2 3 4 5 6 7 8 9 10 11 12 Figure 9.1 Table 9.1 Clock 12 Arithmetic + 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9101112 1 2 3 4 5 6 7 8 9 101112 1 2 3 4 5 6 7 8 9 10 1112 1 2 3 4 5 6 7 8 910 11 12 1 2 3 4 5 6 7 8 91011 12 1 2 3 4 5 6 7 8 9101112 Antoniodiaz/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=