# hw9 - n ∈ N such that q i = 0 for all i ≥ n Find the...

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(1) Let S = (0 , 1) and T = [0 , 1). Let f : S T be given by f ( x ) = x and g : T S be given by g ( x ) = x +1 2 . (a) Find S S ,S T ,S ,T S ,T T ,T (b) give an explicit formula for a 1-1 and onto map h : S T coming from f and g using the proof of the Schroeder-Berenstein theorem. (2) Prove that the set of transcendental numbers is not countable. (3) Prove that the set of functions f : R R has cardinality bigger than R . (4) Let S be the set of sequences q 1 ,q 2 ,q 3 ,... where q i is rational for every i and such that for every sequence there exists
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Unformatted text preview: n ∈ N such that q i = 0 for all i ≥ n . Find the cardinality of S . (5) Show that p 2 + √ 3 and 1 √ 2+ √ 3 are algebraic. Extra Credit: Show that 3 √ 2 + 3 √ 3 is irrational. Hint: First show that x = 3 √ 2 + 3 √ 3 is algebraic by constructing an explicit polynomial f ( x ) with integer coeﬃcients such that f ( x ) = 0. then prove that f ( x ) has no rational roots. 1...
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