546 CHAPTER 9 Mathematical Systems To determine whether a mathematical system is a group under a given operation, check, in the following order, whether (1) the system is closed under the given operation, (2) there is an identity element in the set for the given operation, (3) every element in the set has an inverse element under the given operation, and (4) the associative property holds under the given operation. If any of these four requirements is not met, stop and state that the mathematical system is not a group. If asked to determine whether the mathematical system is a commutative group, you also need to determine whether the commutative property holds for the given operation. Properties of a Commutative Group A mathematical system is a commutative group if all five of the following conditions hold. 1. The set of elements is closed under the given operation. 2. An identity element exists for the set under the given operation. 3. Every element in the set has an inverse element under the given operation . 4. The set of elements is associative under the given operation. 5. The set of elements is commutative under the given operation. Example 1 Whole Numbers Under Addition Determine whether the mathematical system consisting of the set of whole numbers under the operation of addition forms a group. Solution Recall from Chapter 5 that the set of whole numbers is 0, 1, 2, 3, . . . . { } We will check the properties required, using the operation of addition, to determine if the mathematical system is a group. 1. Closure : The sum of any two whole numbers is a whole number. Therefore, the set of whole numbers is closed under the operation of addition. 2. Identity element : The additive identity for the set of whole numbers is 0. For example,1 0 0 1 1, + = + = and20 02 2, + = + = and so on. For any whole number a, a a a 0 0 . + = + = Thus, the mathematical system contains an identity element. 3. Inverse elements : For the mathematical system to be a group, each element must have an additive inverse element in the set . Remember that the additive inverse of a number is the opposite of the number. For example, the additive inverse of 1 is 1, − the additive inverse of 2 is 2, − the additive inverse of 3 is 3, − and so on. Since the numbers − − − 1, 2, 3, … are not in the set of whole numbers, not every number has an inverse in the set. Therefore, this mathematical system is not a group. Because we have already shown that the set of whole numbers under the operation of addition is not a group, there is no need to check the associative property. 7 Now try Exercise 23 Example 2 Rational Numbers Under Multiplication Determine whether the set of rational numbers under the operation of multiplication forms a group. Solution Recall from Chapter 5 that the rational numbers are the set of numbers of the form , p q where p and q are integers and q 0. ≠ The set of rational numbers includes all fractions and integers. 1. Closure : The product of any two rational numbers is a rational number. Therefore, the rational numbers are closed under the operation of multiplication. 2. Identity element : The multiplicative identity element for the set of rational numbers is 1. Note, for example, that 3 1 1 3 3, ⋅ = ⋅ = and 1 1 . 3 8 3 8 3 8 ⋅ = ⋅ = For any rational number a, a a a 1 1 . ⋅ = ⋅ = RECREATIONAL MATH Rubik’s Cube KiyechkaSo/Shutterstock Group theory may be viewed as the study of the algebra of symmetry and transformations (see Section 8.5 for a discussion of transformational geometry). One of the best, and perhaps most entertaining, examples of this view is Rubik’s cube . In 1974, Erno Rubik, a Hungarian teacher of architecture and design, presented the world with his popular puzzle. Each face of the cube is divided into nine squares, and each row and column of each face can rotate. The result is approximately 43 quintillion different arrangements of the colors on the cube. Rubik’s cube is the most popular and best-selling puzzle in human history. Although the popularity of the puzzle peaked during the 1980s, clubs and annual contests are still devoted to solving the puzzle. For more information, visit the website www.rubiks.com.
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