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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 Summary

Countability of sets is analyzed.

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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 ...

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