MAT-3110 Midterm: MATH 3110 App State Spring2009 Test1 answer key
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February 19th, 2009: (15 points) de ne a binary operation on z, as follows x y = xy (for all x, y z). Is associative. (c) show that 1 is an identity for . R = is re exive (since x x) and transitive (since x y and y z implies that x z), but. But 2 r/ 1 since 2 6 1. So r is not an equivalence relation since it is not symmetric. (b) let arb if and only if a b is divisible by 2. Describe the equivalence classes. a b divisible by 2 is the same as a b (mod 2). So the equivalence classes are the even integers (= [0] = {. , 2, 0, 2, 4, . and the odd integers (= [1] = {. Alternatively, notice that a b divisible by 2 means that a and b are o by an even number.
