math403 hw3 - Solutions for HWK 3 Based on the HWKs by my...

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Solutions for HWK 3 Based on the HWKs by my former student JINGYUAN LIU October 10, 2009 Problem A Part 1 Every number r Q can be expressed as r = m n where m Z , n N , and ( m,n ) = 1 . It is clear that r is the root of the polynomial f ( x ) = m - nx , hence r A . So Q A . Part 2 The set P k of polynomials of xed degree, k , with integer coe cients z 0 ,z 1 ,...,z k Z consists of elements (polynomials) of the form z 0 x 0 + z 1 x 1 + ··· + z k x k where z k 6 = 0 . Each polynomial is uniquely identi ed by its integer coe cients so there is a 1 : 1 correspondence between elements of this set, and the set Z k × ( Z \{ 0 } ) . We proved in class that Z is countable; Z \ { 0 } ) is countable as a subset of Z . By a proposition proved in the class the Cartesian product of a nite number of countable sets is countable, so the set Z k × ( Z \{ 0 } ) is countable, therefore the set P k is countable, too. Part 3
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