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Question: prove cantors original result for any nonempty set whether finite...

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Prove Cantor's original result: for any nonempty set (whether finite or infinite), the cardinality of S is strictly less than that of its power set 2s. First show that there is a one-to-one (but not necessarily onto) map g from S to its power set. Next assume that there is a one-to-one and onto function f and show that this assumption leads to a contradiction by defining a new subset of S that cannot possibly be the image of the map f (similar to the diagonalization argument).

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