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# Class Note for MATH 323 at UA

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COURSE
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PAGES
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KARMA
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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
Take home nal Math 323 Section 3 Fall 2008 You are not allowed to discuss these problems with anyone else Also you are not allowed to use any sources besides your text book and your class notes You are allowed to quote theorems from the book but not the results of homework problems which is you use you must prove again All problems carry equal weight Please write up your work clearly starting each problem on a new page and when you turn in the exam please make sure that the problems are in order 1 Given sets A and B Let MA B denote the set of all functions with domain A and co domain B We can de ne a relation R on MA B byfA BgAaBand ng ltgt VD Q B we have fg 1Dl E D Prove or give a counterexample for each of the following statements a R is a re exive relation c R is a symmetric relation b R is a transitive relation Hint Show that ng if and only if VD Q B we have g 1D Q f 1D 2 A set I Q R is called an interval if it has the following property a6Ib Ia c b gt CEI a Show that I can be written as a union of all the closed intervals contained in I 239e I UF fab ab g1 Fef b Show that we can nd a countable either denumerable or nite collection of closed intervals am bn Q I such that I Unan bn Hint Consider cases where I is boundedunbounded and also containsdoes not contain its sup and inf c Show that I can also be written as an intersection of open inter vals containing I 239e I 0 gab ab 21 069 3 Given a E N de ne a sequence recursively by 0 1 and a a n13m a Show that xn E Q for all n b Show that 1 S xn S a for all n c Show that there is a 6 lt 1 such that lxn 7 znl S 91 7 an for all n d Show that xn converges to V5 4 A Q R and A denotes the set of all the accumulation points of A a If y E A and U Q R is an open set containing y show that there are in nitely many distinct points in A U b Show that A 01mm zEA c Using this or otherwise show that A is a closed set 5 Prove or disprove the following a A is nite and U is a open subset of R If A Q U there exists an E gt 0 such that for allx E A Nx6 Q U b P is countable and U is a open subset of R If P Q U there exists an E gt 0 such that for all z E P Nx6 Q U c F is closed and U is a open subset of R If F Q U there exists an E gt 0 such that for all z E F Nx6 Q U d K is compact and U is a open subset of R If K Q U there exists an E gt 0 such that for all z E K Nx6 Q U Extra credit 1 The modi ed Dirichlet function is de ned by 1 7 p e z i 7 1n lowest terms f 96 q q l 0 otherw1se a Show that f is discontinuous at every rational number b Show that f is continuous at every irrational number c Does the limit lin f exist Prove your claim If the limit mal 2 exists7 evaluate the limit

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