Determining if set of functions is countable or uncountable? I have a few sets: A, B, C, D, and E, which are the sets of all functions using the below information.
$$Y = \{1, 2, 3, ..., n\}, n \gt 3 \\
A,\text{ from } \mathbb N \to Y \\
B,\text{ from } \mathbb R \to Y \\
C,\text{ from } Y \to \mathbb N\\
D,\text{ from } Y \to \mathbb R\\
E,\text{ from } Y \to Y$$
So A is the set of all functions from $\mathbb N \to Y$ and so forth.
How do I determine which of the above sets is countably infinite, uncountably infinite, or neither (ie. finite)?
If I were to hazard a guess, I'd say $E$ is finite, because (I think) the range of $Y$ is finite, and a mapping from a finite set to a finite set surely isn't infinite?
 A: For set $A$, consider the following map $F:A \rightarrow \mathscr{P}(\mathbb{N})$ where $\mathscr{P}(\mathbb{N})$ is the power set of the natural numbers:
$$F(f) = f^{-1}[\{1\}]$$
Map $F$ can be shown to be surjection, given a subset $S\subseteq \mathbb{N}$, define: $$f_S: \mathbb{N} \rightarrow Y, \ \ f_S(n) = \begin{cases} 1, & n \in S \\ 2, & n \notin S  \end{cases}$$
so $f_S \in A$ and $F(f_S) = S$. Because we have a surjection from $A$ to $\mathscr{P}(\mathbb{N})$, we conclude $|A| \geq |\mathscr{P}(\mathbb{N})|$ and thus uncountable.
A similiar approach will show that set $B$ is also uncountable.
As for sets $C$, $D$ and $E$, consider the maps: $$H_1: C \rightarrow \mathbb{N}^n, \ \ H_1(f) = (f(1),f(2), \dots,f(n)) \\ H_2: D \rightarrow \mathbb{R}^n, \ \ H_2(f) = (f(1),f(2), \dots,f(n)) \\H_3: E \rightarrow Y^n, \ \ H_3(f) = (f(1),f(2), \dots,f(n))$$
Try to show $H_1,H_2,H_3$ are bijections, and use what you know about the cardinallities of $Y^n, \mathbb{N}^n, \mathbb{R}^n$ to find the cardinalities of $E, C,D$.
