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If X is uncountable and Y is a countable subset of X , show that X \Y has the same cardinality as X , assuming that N has smaller cardinality than X \Y (which cannot be proved in general without the axiom of choice, to be treated in §1.5). Hint: Let B be a countably infinite subset of X \Y . Then B and B ∪ Y have the same cardinality.

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