Question

(1) For each of the following functions, determine if it is injective and determine if it is surjective. Justify your answer. (a) f : R → R, f(x) = 2x + 3. (b) g : R → R 2 , g(x) = (2x, 3x −1). (c) h : R 2 → R, h((x, y)) = x + y + 1. (d) j : {1, 2, 3} → {4, 5, 6}, j(1) = 5, j(2) = 4, j(3) = 6.

(2) Each of the following functions are bijections. Compute their inverses. (a) f : R → R, f(x) = 3x + 2. (b) g : R 2 → R 2 , g((x, y)) = (5x + 4y, 4x + 3y). (c) h : R \ {2} → R \ {5}, h(x) = 5x + 1 x − 2 .

(3) Let f : R → R be defined by f(x) = x 2 . Compute the following images / preimages: (a) f((0, 1)). (b) f([1, 3]). (c) f −1 ([−4, 4]). (d) f −1 ({9}). (e) f −1 ({−9}).

(4) Let f : A → B and suppose C1, C2 ⊆ A. Prove that f(C1 ∪ C2) = f(C1) ∪ f(C2).

(6) Let A = {1, 2, 3, 4, 5}. For each of the following relations on A, determine if it is an equivalence relation or not. If it is an equivalence relation, what are the equivalence classes? If not, 1 why not? P ={(1, 1),(2, 2),(3, 3),(4, 4),(5, 5)}, Q ={(1, 2),(2, 3),(3, 4),(4, 5),(5, 1)}, R ={(1, 1),(2, 2),(2, 4),(3, 3),(3, 5),(4, 2),(4, 4),(5, 3),(5, 5)}, S ={(1, 1),(1, 2),(2, 2),(2, 3),(3, 3),(3, 4),(4, 4),(4, 5),(5, 5)}.

Answer 1

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