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    Real Analysis : Countable Sets and Antichains

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    Answer the following by establishing 1-1(one to one) correspondence with a set of known cardinality:
    1 - Is the set of all functions from{0,1} to N countable or noncountable?
    2 - Is the set of all functions from N to {0,1} countable or noncountable?
    3 - Given a set B ,a subset A of P(B) is called an antichain if no element of A is a subset of any other element of A. Does P(N) contain an uncountable antichain?

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    Solution Preview

    1. countable

    For each of f from {0,1} to N, we consider two cases. If f(0)=f(1), then each such f is corresponding to one element f(0) in N. So it is countable. If f(0)<>f(1), then each such f is corresponding ...

    Solution Summary

    Countability of sets is analyzed.