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Unformatted text preview: Take Home Exam Solutions 1. For each of the following, either &nd an example satisfying the given condi tions or give not long but complete explanation (a short proof quoting some known fact or theorem) of why there is or why there is no such ex ample. (a) An onto function f : N ! R : (b) A bounded set B & R which does not have a largest element. (c) A bounded sequence in R that has no cluster point. (d) An increasing sequence that is not Cauchy. Solution 1 (a) If we had such a map f; we could construct a 11 map g : R ! N . But that would imply that R has the same cardinality as the subset g ( R ) of N , thus would be &nite or countable, a contradiction. There is no such map. (b) The open interval (0 ; 1) is such a set. (c) This is impossible by the BW Theorem. (d) x n = n 2 is such an example. 2. Let f : S ! T and let A & S; B & T: Prove or disprove (by a counterex ample) the following: (a) f & 1 ( f ( A )) = A: (b) f & f & 1 ( B ) ¡ = B: Solution 2 f : R ! R , f ( x ) = x 2 will provide a counterexample for both...
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This note was uploaded on 06/25/2010 for the course MATH MATH 301 taught by Professor Alberterkip during the Fall '08 term at Sabancı University.
 Fall '08
 ALBERTERKIP
 Math

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