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PS2-Solutions - MAT 337 Problem Set 2 Sample Solutions Ian...

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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 Schr¨oder-Bernstein 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¨ oder-Bernstein. We already know that | [0 , 1] | = | R | and [0 , 1] 2 = R 2 , since in both cases the latter is a countable union of copies of the former. This proves | R | = R 2 .
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