Function space notation

Question

Does the function domain notation $f:Q \times Z \rightarrow \mathbb{R}$ denote a binary tuple just like $f: \mathbb{R^{2}} \rightarrow \mathbb{R}$ denotes a tuple like $(\mathbb{R}, \mathbb{R})$?

Answer 1

Yes, it does. In general $\times$ is the Cartesian product between sets given by

$$A\times B=\{(a,b)\mid a\in A, b\in B\}$$

A function $f:A\times B\to C$ is then just a function from this set of tuples, the cartesian product, to $C$.

However, if $(\mathbb R, \mathbb R)$ should represent $\mathbb R^2$, then this is really uncommon notation.

Answer 2

No, $f$ is a function that assigns a real number to each tuple $(q, z) \in Q\times Z$.
$\Bbb R^2\to\Bbb R$ is the special case when $Q=Z=\Bbb R$.