Can I think of functions as a type of mathematical structure?

Question

Consider the function $f:\mathbb{Z} \rightarrow \mathbb{Z} $. Can I think of $f$ as the following 2-tuple $(\mathbb{Z},$ rule to map one integer to another integer$ )? $

And in the above sense, can I think about functions (and in general relations) as a type of mathematical structure?

Answer 1

That is almost correct. You can think of it as the 2-tupple $\langle \mathbb{Z} \times\mathbb{Z}, \text{rule to map one integer to another integer} \rangle $. The domain and codomain of a function can be different.

For example $f(n) = n^2 : \mathbb{Z} \to \mathbb{Z}$ would be $\langle \mathbb{Z} \times\mathbb{Z}, x \mapsto x^2 \rangle$

Answer 2

I am not sure if this is what you are looking for.

In general, given two sets, $A$ and $B$, one can think of a function $f$ from $A$ to $B$ as a subset of $A\times B$. This will be the subset of elements of the form $(a,f(a))$. In the other direction, given a subset $F$ of $A\times B$, one can sometimes find a function corresponding to this subset. This is possible precisely when for every element $a$ of $A$, the set $(\{a\} \times B) \cap F$ consists of exactly one element.