# Is there a standard shorthand notation for typebounds?

Let's assume I want to declare a function. The full "signature" (or "typebound") is given by:

$f(a,b,c)\rightarrow d$ where $a\in \mathbb{N}, b\in \mathbb{Q}, c\in \mathbb{N}, d\in \mathbb{N}$

Is there any generally accepted shorthand notation for the line above, without using $\in$ repeatedly? Something like:

$f:<\mathbb{N},\mathbb{Q},\mathbb{N}>\rightarrow\mathbb{N}$

Also, I would need to use structures (e.g. matrices of certain sizes) in that notation as well, not just predefined sets of numbers (like $\mathbb{N}$ and $\mathbb{Q}$).

• $f: \mathbb{N} \times \mathbb{Q} \times \mathbb{N} \rightarrow \mathbb{N}$ would be something I'd write. E.g. $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $(x,y) \mapsto \sqrt{x^2+y^2}$, you can also put your matrices in there. – Maximilian Gerhardt Jan 12 '16 at 18:08

It's called Cartesian product: if $f$ takes $n$ arguments $x_1, x_2, \dots, x_n$, and $x_i \in X_i \ \forall 1 \le i \le n$, and if $f$ takes values in $Y$, then you write the "signature" of $f$ as $f : X_1 \times X_2 \times \dots \times X_n \to Y$ or, to get rid of the dots, $f : \prod \limits _{i = 1} ^n X_i \to Y$.