544 CHAPTER 9 Mathematical Systems Is there an identity element for the set of integers under the operation of multiplication? The answer is yes; it is the number 1. Note that 2 1 1 2 2, ⋅ = ⋅ = 3 1 1 3 3, ⋅ = ⋅ = and so on. For any integer a, a a a 1 1 . ⋅ = ⋅ = For this reason, 1 is called the multiplicative identity element for the set of integers. Inverse Elements What integer, when added to 4, gives a sum of 0; that is, 4 + 0? = The shaded area is to be filled in with the integer 4: 4 ( 4) 0. − + − = We say that 4− is the additive inverse of 4 and that 4 is the additive inverse of 4. − Note that the sum of the element and its additive inverse gives the additive identity element 0. What is the additive inverse of 12? Since + − = − 12 ( 12) 0, 12 is the additive inverse of 12. Other examples of integers and their additive inverses are Element Additive Inverse Identity Element 0 0 0 3 ( 3) 0 5 5 0 + = + = + − = − + = Note that for the operation of addition, every integer a has a unique inverse, a, − such that a a a a ( ) 0. + − = − + = Is the set of integers closed under the operation of subtraction? Again, the answer is yes. The difference of any two integers is an integer. Is the set of integers closed under the operation of division? The answer is no because two integers may have a quotient that is not an integer. For example, if we select the integers 2 and 3, the quotient of 2 divided by 3 is , 2 3 which is not an integer. Thus, the integers are not closed under the operation of division. We showed that the set of integers was not closed under the operation of division by determining two integers whose quotient was not an integer. A specific example illustrating that a specific property is not true is called a counterexample. Mathematicians and scientists often try to determine a counterexample to confirm that a specific property is not always true. Identity Element Is there an element in the set of integers that, when added to any given integer, results in a sum that is the given integer? The answer is yes. The sum of 0 and any integer is the given integer. For example, 1 0 0 1 1, 4 0 0 ( 4) 4, +=+=−+=+−=− and so on. For this reason, we call 0 the additive identity element for the set of integers. Note that for any integer a, a a a 0 0 . + = + = Definition: Identity Element An identity element is an element in a set such that when a binary operation is performed on it and any given element in the set, the result is the given element. Learning Catalytics Keyword: Angel-SOM-9.1 (See Preface for additional details.) Definition: Inverse Elements When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the inverse of the other element.
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