What is the cardinal number of the following set and how do you prove it? $$ \left \{ f:\mathbb{N}\rightarrow \left \{ 0,1 \right \} :\left|f^{-1}[\left \{ 1\right \}]\right|=\left|f^{-1}[{\left \{ 0 \right \}}]\right| \right \} $$
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Is $|f^{-1}[0]|$ the cardinality of the pre-image of $0$? – Malcolm Dec 21 '17 at 20:43
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I have just edited it, it's the set of all sources of 0. – Ofek Dec 21 '17 at 20:46
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Then the way I am reading the question, asking for functions $|f^{-1}[1]|=|f^{-1}[0]|$ is the same as asking for partitions of $\Bbb{N}$ into two subsets of equal cardinality is the same as asking for the cardinality of the set of infinite subsets of $\Bbb{N}$ is ... – Malcolm Dec 21 '17 at 20:49
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1@Malcolm ... that also have infinite complement. – Dec 21 '17 at 20:50
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@despaigne Yes, thanks. – Malcolm Dec 21 '17 at 20:51
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1@Ofek The number of finite subsets of $\mathbb{N}$ is countable. For the same reason the number of subset of $\mathbb{N}$ that have finite complement is also countable. Therefore the number of infinite subset that also have infinite complement, must be uncountable, as the total number of subset of any kind is uncountable. – Dec 21 '17 at 20:53
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Its cardinality is $2^{\aleph_0}$. For example map elements $3n$ to $0$ and $3n+1$ to $1$. Then for elements, $3n+2$, divide them into two classes, there are $2^{\aleph_0}$ many ways to do this, and map one class to $0$, the other to $1$.
Rene Schipperus
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