# Real Analysis : Countable Sets and Antichains

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.

$2.19