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### How many infinite subsets of N are there anyway? [duplicate]

I was reading 2 proofs one that the powerset of $N$ has a higher cardinality than $N$ two a proof that the cardinality of the set of all finite subsets of $N$ has the same cardinality than $N$ ...
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### The set of all finite subsets of $\mathbb{N}$ is countable [duplicate]

Can someone explain this in the most straightforward manner possible? I've looked at one other thread on here, but the mathematics was unfortunately too confusing for me. Is there some way I can ...
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### prove that $|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$ [duplicate]

I need to prove that: $$|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$$. it's allowed to use the fact that $|P(\mathbb{N})|=2^{\aleph_{0}}=|\mathbb{R}|$ - this is the ...
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### Set of all finite subsets of $\mathbb{N}$ is a countable set [duplicate]

Possible Duplicate: Show that the set of all finite subsets of $\mathbb{N}$ is countable. How can I prove in a proper way that the "set of all finite subsets of $\mathbb{N}$ (the set of natural ...
### Number of one-to-one Function from $\mathbb{N}$ to $\mathbb{N}$ [closed]
How many one-to-one functions are there from $\mathbb{N}$ to $\mathbb{N}$, Where $\mathbb{N}$ is the set of Natural numbers?