smc

Question

Let be the symbol denoting the binary operation on the set S of all non-zero real numbers as follows: For any two numbers a and b of S, a b=2ab. Then the one of the following statements which is not true, is

Accepted Answer

(A) is commutative over S a b = b a = 2ab Statement A is true. (B) is associative over S a (b c) = a 2bc = 2a2bc = 4abc (a b) c = 2ab c = 2(2ab)(c) = 4abc Statement B is true. (C) 12 is an identity element for in S a 12 = a Statement C is true. (D) From the previous answer choice, we know 12 is the identity element for in S. For inverses to exist, they must evaluate to the identity under the operator . Thus, a b = 12, which leads to b = 14a Statement D is true. (E) 12a is an inverse for of the element a of S 12a a = 1 Since the identity is 12, not 1, we can see that statement E is false. Hence, the answer is 12a is an inverse for of the element a of S.