hw2solutions - R a = { z : a z n + a 1 z n-1 + + a n-1 z +...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: January 22, 2010 Assignment 2 1. Prove that the collection F ( N ) of all finite subsets of N is countable. Note that F ( N ) = [ n =1 P ( { 1 , 2 ,...,n } ) . The latter is a countable union of finite sets, as P ( { 1 , 2 ,...,n } ) has 2 n elements. See Exercise 11 in Section 1.3 of the text. 2. A complex number z is said to be algebraic if there are integers a 0 ,...,a n , not all zero, such that a 0 z n + a 1 z n - 1 + ··· + a n - 1 z + a n = 0 . Prove that the set of all algebraic numbers is countable. For fixed a = ( a 0 ,...,a n ), the set
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Unformatted text preview: R a = { z : a z n + a 1 z n-1 + + a n-1 z + a n = 0 } contains at most n elements. Thus, for n xed, [ a =( a ,...,a n ) Z n R a is a countable union of nite sets. Thus it is countable. The countability n-tuples from a countable set was shown in class. Finall, the set of algebraic numbers is [ n N " [ a Z n R n # , which in tern is a countable union of countable sets and is thus countable. 1...
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