Intrepreting tuples as functions I have been mulling over this for a while now. I am told $\mathbb R^n$ can be interpreted as a set of functions.
Take $\mathbb R^2$, for example I can see how we might interpret it as a set containing all ordered pairs: $<x_1,x_2>$. However I do not understand the notation:
$  \{ f:\{1,2\} \longrightarrow \mathbb R \} = \mathbb R^2$
This would mean we have a domain with two elements and a codomain with $| \mathbb R|$ elements(which doesn't make sense to me). What would the values of $f(1)$ and $f(2)$ be by this definition?
Basically I'm asking what justification is there for the following:

$$  \{ f:\{1,2\} \longrightarrow \mathbb R \} = \{\langle x_1,x_2\rangle:x_1,x_2 \in \mathbb R^2\}$$

For instance it's easy to see why the R.H.S. can be interpreted as Cartesian plane but how does two-dimensional plane relate to the L.H.S in the above?

Consider the following sets:
$$A:=\{(a,b):a,b\in\Bbb R\}, \\
 B:=\{f:\{0,1\}\to \Bbb R\}$$
An element of A looks like $(x,y)$, an element of B is a function $f:0\mapsto r,1\mapsto t; r,t\in \Bbb R$.
We have the natural bijection $F((x,y))=f:0\mapsto x, 1\mapsto y$, so we might as well treat these sets as equals.
