# How to determine whether the following sets are countable:

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

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