Question

Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every function f from R to 2R, there is a real number x such that f-?(x) Finish the proof by showing how to use ß to make a function from R to 10, 1 that is not in the range of ß, which contradicts the assumption that B is a l-to-1 and onto function,

Answer 1

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