9.2 Finite Mathematical Systems 555 Table 9.9 $ % @ $ $ @ % % @ % $ @ % $ @ Example 5 Another Finite Mathematical System Determine whether the mathematical system defined by Table 9.9 is a commutative group under the operation of . Solution 1. The system is closed. 2. No row is identical to the top row, so there is no identity element. Therefore, this mathematical system is not a group. There is no need to go any further, but for practice, let’s look at a few more items. 3. Since there is no identity element, there can be no inverses. 4. The associative property does not hold. The following counterexample illustrates the associative property does not hold for every case. ($%)@ $(%@) @ @ $ $ @ $ ? ? = = ≠ 5. The table is symmetric about the main diagonal. Therefore, the commutative property does hold for the operation of . Note that the associative property does not hold even though the commutative property does hold. This outcome can occur when there is no identity element and not every element has an inverse, as in this example. 7 Now try Exercise 59 Did You Know? Creating Patterns by Design A A (b) (a) (c) (d) A A A A A A Patterns from Symmetries of Culture by Dorothy K. Washburn and Donald W. Crowe (University of Washington Press, 1988) m Patterns, by the Pueblo people of New Mexico, from Symmetries of Culture by Dorothy K. Washburn and Donald W. Crowe (University of Washington Press, 1988) What makes group theory such a powerful tool is that it can be used to reveal the underlying structure of just about any physical phenomenon that involves symmetry and patterning, such as wallpaper or quilt patterns. Interest in the formal study of symmetry in design came out of the Industrial Revolution in the late nineteenth century. The new machines of the Industrial Revolution could vary any given pattern almost indefinitely. Designers needed a way to describe and manipulate patterns systematically. Shown here are the four geometric motions that generate all two-dimensional patterns: (a) reflection, (b) translation, (c) rotation, and (d) glide reflection. How these motions are applied, or not applied, is the basis of pattern analysis. The geometric motions, called rigid motions , were discussed in Section 8.5. Example 6 Is the System a Commutative Group? Determine whether the mathematical system defined by Table 9.10 is a commutative group under the operation of .▲ Table 9.10 ▲ A B C A A B C B B B A C C A C Solution 1. The system is closed. 2. There is an identity element, A. 3. Each element has an inverse element; A is its own inverse, and B and C are inverses of each other. 4. Every element in the set does not appear in every row and every column of the table, so we need to check the associative property carefully. There are many specific cases in which the associative property does hold. However, the following counterexample illustrates that the associative property does not hold for every case. ▲ ▲ ▲ ▲ ▲ ▲ B B C B B C B C B A A B ( ) ( ) ? ? = = ≠ 5. The commutative property holds because there is symmetry about the main diagonal. Since we have shown that the associative property does not hold under the operation of ,▲ this system is not a group. Therefore, it cannot be a commutative group. 7 Now try Exercise 61
RkJQdWJsaXNoZXIy NjM5ODQ=