What is the difference between those expressions?

Question

I just saw this mapping in my course and can't understand what it means?

It says "F(E,E) is the set of mappings of E in E defined by:"

What is the difference with this? $$E \times E \rightarrow E$$

Thank you so much

Actually, I can't really understand the definition? How can you define something ($F(E,E)$) by using itself ($F(E,E)$ x $F(E,E)$ -> $F(E,E)$) to define itself ? Is this some sort of recursive definition ?

Answer 1

Usually a writing with that arrow denotes the pre-image and image of a function. You have ${\cal F}(E,E)$ as all mappings from some set $E$ into the same set $E$. However if you (for example) consider the composition of two of those function, the composition itself is a function, that takes two mappings and give you a result of a mapping.

In Short: The function, whose pre-image and image are defined here, is not such a mapping from ${\cal F}$ but something that uses these mappings as input and output.

Answer 2

$\mathcal{F}(E,E) \times \mathcal{F}(E,E)=\{(f,g):f,g \in \mathcal{F}(E,E)$, where $f,g: E \to E$

A mapping $\phi:\mathcal{F}(E,E) \times \mathcal{F}(E,E) \to \mathcal{F}(E,E)$ maps a pair $(f,g)$ to a map $\phi(f,g) \in \mathcal{F}(E,E)$ .