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How one can embedding $P(\mathbb{N})$ into $P_{\infty}(\mathbb{N})$, where $P_{\infty}(\mathbb{N})=\{A\in P(\mathbb{N})\ \ : \ \ A \ \ \ \text{is infinit set} \}$?

I've tried defining a following function, $f:P(\mathbb{N})\to P_{\infty}(\mathbb{N}) $, as:

$f(A)=\mathbb{N}\setminus A$

But there is a problem when $A=\mathbb{N}$

Thank you.

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    It is a power set of $\mathbb{N}$2012-11-28

1 Answers 1

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Define the following function.

$f(A)=\{k\in\Bbb N\mid k\text{ is odd}\}\cup\{2k\mid k\in A\}$

It is easy to show that $f(A)$ is always infinite. Now you have to show that it is also injective, which I am trusting you can do on your own.


The idea is to use the fact that we can split $\Bbb N$ to two disjoint parts both of the same cardinality as $\Bbb N$. Now fix some bijection between $\Bbb N$ and one part and use that to send subsets into subsets of the one part, union with the second part. The result will always be infinite, and it is not difficult to show that it is indeed an injective function.

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    Great! Thanks a lot!2012-11-28