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Unformatted text preview: MAT 337 Problem Set 2 Sample Solutions Ian Zwiers  February 10, 2009 This document is for demonstration purposes; it is not a marking key. 2.9 (E) Show R 2 =  R  . It is easy to define a surjective function f : [0 , 1] 2 [0 , 1]; for example f ( x,y ) = x . We now construct a surjective function g : [0 , 1] [0 , 1] 2 . For each z [0 , 1], let ( z n ) n =1 denote a decimal expansion. Define g ( z ) = ( z odd ,z even ) where z odd has decimal expansion ( z 1 ,z 3 ,z 5 ,... ) and z even has decimal expansion ( z 2 ,z 4 ,z 6 ,... ). This proves [0 , 1] and [0 , 1] 2 have the same cardinality. I have used a version of SchroderBernstein that might not appear in the textbook. Technically, we create a function [0 , 1] [0 , 1] 2 by trimming f 1 ; this new function is necessarily injec tive. Note that f 1 is NOT itself a function. Repeat with g 1 and apply Schr oderBernstein....
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This document was uploaded on 03/27/2010.
 Spring '09

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