1.37 THEOREM. Suppose that A and B are sets. (i) If A ~ Band B is at...
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1.37 THEOREM. Suppose that A and B are sets.
(i) If A ~ Band B is at most countable, then A is at most countable.
(ii) If A ~ B and A is uncountable, then B is uncountable.
(iii) R is uncountable.
PROOF. (i) Since B is at most countable, choose by Lemma 1.36 a function 9 that takes N onto B. We may suppose that A is nonempty, hence fix an ao E A.
Then f(n) := { g(n)
ao g(n) E A g(n) tt A
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