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How to determine whether the following sets are countable:

i.collection of all finite subsets of $\mathbb N$

ii.the collection of all functions from $\mathbb N$ to $\mathbb R$

iii.collection of all roots of polynomials in single variable over $\mathbb Z$

I am reading set theory but I cant determine how to construct the functions from the above sets to $\mathbb N$ or whether they exist

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  1. How many subsets of $\mathbb N$ of $1$ element exist? How namy of size $n$, for an arbitrary $n$? What could you say aabout all them together?

  2. For every real number, take the function that is constantly that real number. How many of those functions have you found?

  3. How many polynomials of degree $n$ exist with integer coefficients? How many roots any of those polynomial has?

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