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Adding another. Suppose that you have any infinite set (it could be really huge), and you wish to add one new element to the set. Prove that this new set (the old set with the new element thrown in) has the same cardinality as the original set. (Hint: The infinite set has a subset that is in one-to-one correspondence with the natural numbers. Use previous ideas to add the new element to this subset, and let the rest of the set correspond with itself.)
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