MAT-3110 Midterm: MATH 3110 App State Spring2010 Test1 answer key
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Be sure to show your work! (10 points) let s = r {0} (the set of non-zero reals). For x, y s, de ne x y = x 1y. Suppose that x, y s. this means that x and y are non-zero real numbers. Thus x 1 = 1/x is a non-zero real number and so x y = x 1y is a non-zero real number (a non-zero real times a non-zero real is a non-zero real). Thus x y s. (b) the operation is not associative. To show fails to be associative, we need to nd a single example where associativity does not work. Let"s try (somewhat randomly) 3, 2, 1 s. then (3 2) 1 = (3 12) 1 = (2/3) 1 = (2/3) 11 = 3/2 whereas. 3 (2 1) = 3 (2 11) = 3 (1/2) = 3 1(1/2) = 1/6. Thus (3 2) 1 6= 3 (2 1).
