How to determine if these sets involving rationals are countable? I am wondering whether the set of polynomial functions from $\mathbb{Q} \to \mathbb{Q}$ is countable, whether the set of maps from $\mathbb{Q} \to \mathbb{Q}$ with finite image is countable, and whether the set of finite subsets of $\mathbb{Q}$ is countable. 
For the first question, I tried to approach the problem by considering that there are only finitely many orders of polynomials (the integers), but got tripped up once I had to consider coefficients. In general, I feel like there is an easier way to determine this than I am envisioning at the moment and would appreciate any help.  
 A: Your approach to the first question is the right start. For a degree $n$ polynomial, we have $n+1$ coefficients: $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$. So we can view a polynomial as an $n$-tuple $(a_n, \ldots, a_0)$ of rationals. Thus degree $n$ polynomials are essentially just elements of $\mathbb{Q}^{n+1}$, more precisely these sets are in bijection. Then the set of all polynomials (on the rational numbers) is in bijection with the set
$$
\bigcup_{n \in \mathbb{N}} \mathbb{Q}^{n+1}.
$$
This is the approach that was mentioned by fleablood in the comments.
For the second question, Don Thousand already made a comment that the functions $\mathbb{Q} \to \{0,1\}$ are in bijection with $\mathcal{P}(\mathbb{Q})$, the powerset of $\mathbb{Q}$, which is uncountable. Since these are definitely contained in the set of all functions with finite image, that set must be uncountable.
Finally, the set of finite subsets of a countable set is again countable. The trick to prove this is similar to the first question. Given any finite tuple $(a_1, \ldots, a_n)$ we can form a finite set $\{a_1, \ldots, a_n\}$ (note that this set may have less than $n$ elements, because the tuple may contain duplicates). This way we obtain a surjection
$$
\bigcup_{n \in \mathbb{N}} \mathbb{Q}^{n+1} \to \mathcal{P}_{\text{fin}}(\mathbb{Q}),
$$
where $\mathcal{P}_{\text{fin}}(\mathbb{Q})$ denotes the set of finite subsets of $\mathbb{Q}$. In the first question we established that $\bigcup_{n \in \mathbb{N}} \mathbb{Q}^{n+1}$ is countable, so $\mathcal{P}_{\text{fin}}(\mathbb{Q})$ must indeed be countable.
Edit: Don Thousand's comment I referred to is removed, but still I'd like to keep the reference because I liked that elegant insight.
