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MATH 409.501 Examination 1 February 15, 2012 Name: ID#: The exam consists of 3 questions. The point value for a question is written next to the question number. There is a total of 100 points. No aids are permitted. 1. [40] In each of the following eight cases, indicate whether the given statement is true or false. No justification is necessary. (a) Let E be a set and suppose that there exists a surjective function f : R → E. Then E is uncountable. (b) If E is a subset of R which has a supremum, then the set −E = {−x : x ∈ E} has an infimum. (c) Let a ∈ R. Then |a| < ε for all ε > 0 if and only if a = 0. 1 (d) If {Ex }x∈R is a collection of finite sets indexed by the real numbers, then is at most countable. S x∈R Ex (e) Every subset of R has at most two suprema. (f) Let f : R → R be defined by f (x) = x2 . Then f −1 ([0, 1]) = [−1, 1]. (g) Let A1 , A2 , A3 , . . . be nonempty finite subsets of N such that An ∩ Am = ∅ for all distinct n, m ∈ N. Define the function f : N → N by declaring f (n) to be the least element of An . Then f is injective. (h) Let A1 , A2 , A3 , . . . be nonempty bounded subsets of R such that An ∩ Am = ∅ for all distinct n, m ∈ N. Define the function f : N → R by f (n) = sup An . Then f is injective. 2 2. [30] (a) State the well-ordering principle. (b) Prove that 2n−1 ≤ n! for all n ∈ N. 3 3. [30] (a) State the completeness axiom for R. (b) Let A be a nonempty bounded subset of R, and consider the set B = {x2 : x ∈ A}. Prove that sup B exists. (c) Give an example to show that the equality sup B = (sup A)2 may fail in part (b). 4