MAT 102 - ASSIGNMENT # 2 DUE OCTOBER 5 (5) Let F be a field. Suppose that 1 + 1 + 1 = 0. Prove that for all x E F, we have x + x + x = 0. (6) Let F = {0, 1, x} be a field with three elements. (a) Complete the following addition and multiplication tables. Explain how you filled in each missing entry. + 01x . 0 1x 0 0 1 x (b) Find the value of x2 + x + 1.Assigned Questions (1) Let f(x) = 3x - 1 We can consider this formula as defining a function with co-domain R. ac + 2 However, the expression does not make sense for all x E R, so the domain of f is not R. (a) What is the maximal subset D C R for which this expression defines a function f : D - R? (b) Let f : D - R be the function given by f () =- 3x - 1 x + 2 , where D is the set in (a). Find the range (or image) of f. Prove your claim. (Do not use calculus!) (2) Let f : N - Q be the function defined by f(x) _ 23 + 6x2 + 4x 3 (a) Factor the numerator completely. (b) Explain why given three consecutive integers a, a + 1, a + 2, one of them is divisible by 3. (c) Prove that f(x) E N for all x E N. (d) Is the image (or range) of f equal to N? Justify your claim. (3) Let F = {0, 1, a, b} be a field with 4 elements. Given that ab = 1, prove that a2 = b. (4) Let F be a field. Let a, b E F. Prove that if a . b = 0, then either a = 0 or b = 0