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Dr. A. Betten Fall 2009 MATH 281A1 Introduction to Mathematical Reasoning Assignment # 2 Problem # 5 a) Let A and B be sets. Prove that A ∪ B = A if and only if B ⊆ A. b) Prove that (A \ C) ∩ (B \ C) = (A ∩ B) \ C for all sets A, B, C. Problem # 6 Which if the following statements are true and which are false? Give proofs or counterexamples. a) For any sets A, B, C we have A ∪ (C ∩ C) = (A ∪ B) ∩ (A ∪ C). b) For any sets A, B, C we have (A \ B) \ C = A − (B \ C). c) For any sets A, B, C we have (A \ B) ∪ (B \ C) ∪ (C \ A) = A ∪ B ∪ C. Problem # 7 ∞ Work out ∪∞ n=1 An and ∩n=1 An for the following sets a) An = {x ∈ R | x > n} √ b) An = {x ∈ R | n1 < x < 2 + n1 } c) An = {x ∈ R | −n < x < n1 } √ √ d) An = {x ∈ R | 2 − n1 < x < 2 + n1 } Problem # 8 Book 5.19 page 49 Problem # 9 Book 5.20 page 49