9.1 Groups 547 3. Inverse elements : For the mathematical system to be a group under the operation of multiplication, each and every rational number must have a multiplicative inverse element in the set of rational numbers. Remember that for the operation of multiplication, the product of a number and its inverse must give the multiplicative identity element, 1. Let’s check a few rational numbers: Rational Number Inverse Identify Element 7 1 7 1 2 3 3 2 1 1 5 5 1 ⋅ = ⋅ = ⋅ = − ⋅ − = Looking at these examples you might deduce that each rational number does have an inverse. However, one rational number, 0, does not have an inverse. 0 ? 1 ⋅ = Because there is no rational number that when multiplied by 0 gives 1, 0 does not have a multiplicative inverse. Since not every rational number has an inverse, this mathematical system is not a group. There is no need at this point to check the associative property because we have already shown that the mathematical system of rational numbers under the operation of multiplication is not a group. 7 Now try Exercise 25 Timely Tip Identity Element and Inverse Elements When checking to see if a mathematical system is a group, students often confuse the identity element and the inverse elements. Remember, there can be only one identity element in the set, but each element in the set must have an inverse. For example, consider the integers under addition. The additive identity element is the number 0. The additive inverse of every integer is its opposite. For example, the additive inverse of 1 is 1, − the additive inverse of 2 is −2, and so on. In a group, there is only one identity but each element has its own inverse. Example 3 Real Numbers Under Addition Determine whether the set of real numbers under the operation of addition forms a commutative group. Solution Recall from Section 5.5 that the real numbers can be thought of as the numbers that correspond to each point on the real number line. The set of real numbers includes all rational numbers and all irrational numbers. 1. Closure : The sum of any two real numbers is a real number. Therefore, the real numbers are closed under the operation of addition. 2. Identity element : The additive identity element for the set of real numbers is 0. For example, 0 0 2 3 2 3 2 3 + = + = and500 5 5. + = + = For any real number a a a a , 0 0 . + = + = 3. Inverse elements : For the set of real numbers under the operation of addition to be a commutative group, each and every real number must have an additive inverse in the set of real numbers. For the operation of addition, the sum of a real number and its additive inverse, or opposite , must be the additive identity element, 0. From Section 5.5, we saw that every real number a has an opposite a− such that a a a a ( ) 0. + − = − + = Recall also that 0 0 0; + = therefore, 0 is its own additive inverse. Thus, all real numbers have an additive inverse. 4. Associative property : From Section 5.5, we know that the real numbers are associative under the operation of addition. That is, for any real numbers a b , , and c a b c a b c , ( ) ( ). + + = + + 5. Commutative property : From Section 5.5, we know that the real numbers are commutative under the operation of addition. That is, for any real numbers a and b, a b b a. + = + Since all five properties (closure, identity element, inverse elements, associative property, and commutative property) hold, the set of real numbers under the operation of addition forms a commutative group. 7 Now try Exercise 27 Instructor Resources for Section 9.1 in MyLab Math • Objective-Level Videos 9.1 • PowerPoint Lecture Slides 9.1 • MyLab Exercises and Assignments 9.1
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