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Countable/Uncountable Sets

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Please see the attached image for questions I and II.

III) If A is a countable subset of an uncountable set X, prove that X A is uncountable.

IV) Suppose that f is a function from X into Y so that the range of f is uncountable. Prove that X is uncountable.

V) Prove that the set of all polynomials with rational coefficients is countable.

VI) A real number a is said to be an algebraic number if there exists a polynomial p(x) with integer coefficients such that p(a) = 0. Prove that the set of algebraic numbers is countable. (Hint: use V))

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Countable/Uncountable Sets are clarified.

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