This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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 1-1 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 B-W 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...
View Full Document
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