1 A set is denumerable if it is equivalent to N.docx - 1 A...

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1 A set is denumerable if it is equivalent to N . A set is countable if it is finite or denumerable. If a set is denumerable, we say it has cardinality d or cardinal number d 8 . Thus a set is denumerable iff it its members can be enumerated in a (nonterminating) sequence ( a 1 ,a 2 ,...,a n ,... ). We show below that this fails to hold for infinite sets in general. The following may not seem surprising but it still needs to be proved. Theorem 4.6.2 Any denumerable set is infinite (i.e. is not finite). Proof: It is sufficient to show that N is not finite (why?). But in fact any finite subset of N is bounded, whereas we know that N is not (Chapter 3). We have seen that the set of even integers is denumerable (and similarly for the set

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