9.1 Groups 543 Is the set of integers closed under the operation of multiplication? The answer is yes. When any two integers are multiplied, the product will be an integer. The integers are commutative under the operations of addition and multiplication. For example, Addition Multiplication 2 4 4 2 ? + = + and 2 4 4 2 ? ⋅ = ⋅ 6 6 = 8 8 = The integers, however, are not commutative under the operations of subtraction and division. For example, Subtraction Division 4 2 2 4 ? − = − and 4 2 2 4 ? ÷ = ÷ 2 2 ≠ − 2 1 2 ≠ The integers are associative under the operations of addition and multiplication. For example, Addition Multiplication (12)3 1(23) ? + + = + + and (12)3 1(23) ? ⋅ ⋅ = ⋅ ⋅ 3 3 1 5 ? + = + 2 3 1 6 ? ⋅ = ⋅ 6 6 = 6 6 = The integers, however, are not associative under the operations of subtraction and division. See Exercises 17 and 18. To say that a set of elements is commutative under a given operation means that the commutative property holds for any elements a and b in the set. Similarly, to say that a set of elements is associative under a given operation means that the associative property holds for any elements a b , , and c in the set. Consider the mathematical system consisting of the set of integers under the operation of addition. Because the set of integers is infinite, this mathematical system is an example of an infinite mathematical system. We will study certain properties of this mathematical system. The first property we will examine is closure. Closure The sum of any two integers is an integer. Therefore, the set of integers is said to be closed, or to satisfy the closure property, under the operation of addition. Commutative and Associative Properties For any elements a b , , and c Addition Multiplication Commutative property a b b a + = + a b b a ⋅ = ⋅ Associative property a b c a b c ( ) ( ) + + = + + a b c a b c ( ) ( ) ⋅ ⋅ = ⋅ ⋅ Definition: Closure If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed (or has closure) under the given binary operation.
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