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Math 220 Assignment 4 Due October 9th Definition 1. LetSX and U be sets. For each x ∈ X, let Ax ⊆ U . Define the X-indexed union Ax = {u ∈ U : ∃x ∈ X 3 u ∈ Ax }. Similarly, define the x∈X T X-indexed intersection Ax = {u ∈ U : ∀x ∈ X, u ∈ Ax }. x∈X 1. Let U be the set of positive integers between 1 and 50, inclusive. For an integer d, let Ad be the multiples of d. Let X = {2, 3, 5}. T (a) What is Ax ? x∈X (b) Let P be the set of prime numbers. What is U \ (P ∪ S Ax )? Hint: x∈X it has exactly two elements. 2. Which of the following functions are equal? (a) f : R \ {1} → R given by f (x) = x3 −x x−1 . (b) g : R → R given by g(x) = x2 + x. (c) h : R \ {1} → R given by h(x) = x2 + x. (d) k : R → [−1, ∞) given by k(x) = x3 −x x−1 . 3. Let U = Z, and let Z≥2 denote the set of positive integers greater than or equal to 2. For d ∈ Z+ , let Qd = {x ∈ Q : dx ∈ Z}. (a) Prove that Q2 ∩ Q3 = Z. S (b) Prove that Qd = Q. d∈Z≥2 4. Let U and X be sets. Let A, B, C be subsets of U , and for each x ∈ X, let Ax ⊆ U . Prove or disprove: (a) U \ (A ∩ B ∩ C) = (U \ A) ∪ (U \ B) ∪ (U \ C). T S (b) U \ ( Ax ) = (U \ Ax ). x∈X x∈X 1 5. Let O be the set of odd integers, and E be the set of even integers. Define C : Z → Z by n n∈E 2 C(n) = 3n + 1 n ∈ O (a) Prove that C(E) = Z. (b) Prove that C(O) = {n ∈ Z : n%6 = 4}. (c) Prove that C −1 (O) = {n ∈ Z : n%4 = 2}. 2