552 CHAPTER 9 Mathematical Systems One method that can be used to determine whether a system defined by a table is commutative under the given operation is to determine whether the elements in the table are symmetric about the main diagonal . The main diagonal is the diagonal from the upper left-hand corner to the lower right-hand corner of the table. In Table 9.3, the main diagonal is shaded in pink. If the elements are symmetric about the main diagonal, then the system is commutative. If the elements are not symmetric about the main diagonal, then the system is not commutative. If you examine the system in Table 9.3, you see that its elements are symmetric about the main diagonal because the same numbers appear in the same relative positions on opposite sides of the main diagonal. Therefore, this mathematical system is commutative. It is possible to have groups that are not commutative. Such groups are called noncommutative or nonabelian groups . However, a noncommutative group must have at least six distinct elements . Nonabelian groups are illustrated in Exercises 67 through 69. Now we will look at another finite mathematical system. 4. Associative property : Now consider the associative property. Does a b c ( ) + + = a b c ( ) + + for all values a b , , and c of the set? Remember to always evaluate the values within the parentheses first. Let’s select some values for a b , , and c. Let a b 2, 6, = = and c 8. = Then + + = + + + = + = (26)8 2(68) 8 8 2 2 4 4 ? ? True Let a b 5, 12, = = and c 9. = Then (5 12) 9 5 (12 9) 5 9 5 9 2 2 ? ? + + = + + + = + = True Randomly selecting any elements a b , , and c of the set reveals a b c ( ) + + = a b c ( ). + + Thus, the system of clock 12 arithmetic is associative under the operation of addition. Note that if there is just one set of values a b , , and c such that a b c a b c ( ) ( ), + + ≠ + + the system is not associative. Usually, you will not be asked to check every case to determine whether the associative property holds. However, if there is a row or a column that does not contain every element in the system, then the associative property must be checked carefully. 5. Commutative property : Does the commutative property hold under the given operation? Does a b b a + = + for all elements a and b of the set? Let’s randomly select some values for a and b to determine whether the commutative property appears to hold. Let a 5 = and b 8; = then Table 9.1 shows that 5 8 8 5 1 1 ? + = + = True Let a 9 = and b 6; = then 9 6 6 9 3 3 ? + = + = True The commutative property holds for these two specific cases. In fact, if we were to select any values for a and b, we would find that a b b a. + = + Thus, the commutative property of addition is true in clock 12 arithmetic. Note that if there is just one set of values a and b such that a b b a, + ≠ + the system is not commutative. This system satisfies the five properties required for a mathematical system to be a commutative group. Thus, clock 12 arithmetic under the operation of addition is a commutative, or abelian, group. 7 Now try Exercise 43 RECREATIONAL MATH Permutation Puzzles In addition to Rubik’s cube ( see Recreational Math box on page 546), group theory is used in many other puzzles known as permutation puzzles. Many permutation puzzles are variants of Rubik’s cube, including a popular puzzle called Rubik’s 360. Another example of a permutation puzzle is the classic sliding tile puzzle called the 15-puzzle, shown above. The relationship between permutation puzzles and group theory is explored in detail in the book Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys , by David Joyner. Table 9.3 Symmetry About the Main Diagonal + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 Garry518/123RF
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