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I often throw around signatures like:

$$f: X \times X \to X$$

and

$$f: (x_1, x_2) \mapsto x_1 \cdot x_2$$

But how do you call these kinds of signatures?

EDIT: Some clarification:

  1. The first version maps sets and uses $\to$
  2. The second version maps variables and uses $\mapsto$

How do you call these different kinds of notations, because you usually write down both for a specific function (one to show domain and codomain, second to show the graph)

hgiesel
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  • Not very clear... – Mauro ALLEGRANZA Apr 20 '17 at 14:48
  • The symbol $f : X \times X \to X$ means that $f$ is a function from the cartesian product of the set $X$ with itself into the set $X$. Example : $+ : \mathbb N \times \mathbb N \to \mathbb N$ is the sum of natural numbers. It maps every pair $(n,m)$ of naturals into their sum $n+m$. – Mauro ALLEGRANZA Apr 20 '17 at 14:50
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    The second line doesn't make much sense to me; I would write it as $$f: (x_1,x_2)\mapsto x_1+x_2.$$ – user7530 Apr 20 '17 at 15:23
  • I think of $f:X\times X\to X$ as the same kind of thing as $x\in X$, and I don't know a specific name for that either. –  Apr 20 '17 at 15:27

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