9.2 Finite Mathematical Systems 559 In Exercises 67 and 68 the tables shown below are examples of noncommutative, or nonabelian, groups. For each exercise, do the following. a) Show that the system under the given operation is a group. (It would be very time consuming to prove that the associative property holds, but give some examples to show that it appears to hold.) b) Determine a counterexample to show that the commutative property does not hold. 67. As shown in Fig. 9.2, R replaces 1 with 3 and S replaces 3 with 3 (no change), so R S * replaces 1 with 3. R replaces 2 with 1 and S replaces 1 with 2, so R S * replaces 2 with 2 (no change). R replaces 3 with 2 and S replaces 2 with 1, so R S * replaces 3 with 1. R S * replaces 1 with 3, 2 with 2, and 3 with 1. R S 1 3 3 2 1 2 3 2 1 R * S R * S 5 1 2 3 3 2 1 5 T Q R Figure 9.2 Since this result is the same as replacement set T, we write R S T * . = a) Complete the table for the operation using the procedure outlined. * ? 1 2 3 4 5 6 1 5 3 4 2 6 1 2 4 6 5 1 3 2 3 2 1 6 5 4 3 4 3 5 1 6 2 4 5 6 4 2 3 1 5 6 1 2 3 4 5 6 * 68. A B C D E F A E C D B F A B D F E A C B C B A F E D C D C E A F B D E F D B C A E F A B C D E F * Challenge Problems/Group Activities 69. Book Arrangements—A Nonabelian Group Suppose that three books numbered 1, 2, and 3 are placed next to one another on a shelf. If we remove volume 3 and place it before volume 1, the new order of books is 3, 1, 2. Let’s call this replacement R. We can write R 1 2 3 3 1 2 = ⎛ ⎝⎜ ⎞ ⎠⎟ which indicates the books were switched in order from 1, 2, 3 to 3, 1, 2. Other possible replacements are S T U V , , , ,and I, as indicated. S T U V I 1 2 3 2 1 3 1 2 3 3 2 1 1 2 3 1 3 2 1 2 3 2 3 1 1 2 3 1 2 3 = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝⎜ ⎞ ⎠⎟ Replacement set I indicates that the books were removed from the shelves and placed back in their original order. Consider the mathematical system with the set of elements R S T U V I { , , , , , } with the operation of *. To evaluate R S * , write R S R S * * 1 2 3 3 1 2 * 1 2 3 2 1 3 = ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ * R S T U V I R T S T U V I b) Is this mathematical system a group? Explain. * c) Is this mathematical system a commutative group? Explain. No, it is not commutative. For example, R S S R * * . ≠ Recreational Mathematics 70. A Tiny Group Consider the mathematical system with the single element {1} under the operation of multiplication. a) Is this system closed? Yes b) Is there an identity element in the set? Yes; 1 c) Does 1 have an inverse? Yes; 1 d) Does the associative property hold? Yes e) Does the commutative property hold? Yes f) Is {1} under the operation of multiplication a commutative group? Yes Research Activity 71. Twenty-Four-Hour Clock System Do research on the 24-hour clock system (see the Did You Know? box on page 551). Write a paper on its use throughout the world, including in the U.S. military and in medical facilities. *See Instructor Answer Appendix
RkJQdWJsaXNoZXIy NjM5ODQ=