Solution verfication and two small cardinality questions

Question

I'm studying to my final exam due to tomorrow, and I encountered several small problems.

Determine the cardinality of the following sets:

1). $A$ is the set of all injective functions from $\{1,2,3\}$ to $\mathbb R$.

My solution: $1$ can be sent to everything in $\mathbb R$. Lets assume it was sent to $\{a\}$. So $2$ can be sent to $\mathbb R \setminus \{a\} \sim \mathbb R$. Lets assume it was sent to $\{b\}$. $3$ can be sent to $\mathbb R \setminus \{a, b\} \sim \mathbb R$. So overall there are $|\mathbb R \times \mathbb R \times \mathbb R|=\aleph$ such functions.

2). I wasn't sure how to solve this. $B$ is the set of all surjective functions from $\mathbb R$ onto $\{1,2,3\}$.

3)> I tried continuing, but also got stuck on this one: $C= \{f \in B: $ for every $x,y \in \mathbb R| x \leq y \rightarrow f(x)\leq f(y) \}$

Thanks in advance for any hints or assistance of any sort!

Answer 1

The solution to (1) is correct.

Here is a hint to (2) For every subset $A$ of $\Bbb R\setminus\{0\}$ find a surjection from $\Bbb R\setminus\{0\}$ onto $\{1,2\}$ and map $0$ to $3$.

Answer 2

For $B$, I guess it would be $3^{\mathbb{R}}$ as there are 3 choices for every $x\in\mathbb{R}$, and the non-surjective functions are a tiny fraction of that.

For $C$, there are exactly two values where $f(x)$ changes value, so $\mathbb{R}^2$.

I don't know what the various infinite numbers mean; I guess you can convert .