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* the binary operation

Source: 1971 AHSME Problem 6

April 18, 2014
AMC

Problem Statement

Let ∗\ast be the symbol denoting the binary operation on the set SS of all non-zero real numbers as follows: For any two numbers aa and bb of SS, a∗b=2aba\ast b=2ab. Then the one of the following statements which is not true, is
<spanclass=′latex−bold′>(A)</span>∗ is commutative over S<spanclass=′latex−bold′>(B)</span>∗ is associative over S<span class='latex-bold'>(A) </span>\ast\text{ is commutative over }S \qquad<span class='latex-bold'>(B) </span>\ast\text{ is associative over }S\qquad
<spanclass=′latex−bold′>(C)</span>12 is an identity element for ∗ in S<spanclass=′latex−bold′>(D)</span>Every element of S has an inverse for ∗<span class='latex-bold'>(C) </span>\frac{1}{2}\text{ is an identity element for }\ast\text{ in }S\qquad<span class='latex-bold'>(D) </span>\text{Every element of }S\text{ has an inverse for }\ast\qquad
<spanclass=′latex−bold′>(E)</span>12a is an inverse for ∗ of the element a of S<span class='latex-bold'>(E) </span>\dfrac{1}{2a}\text{ is an inverse for }\ast\text{ of the element }a\text{ of }S
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