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?

Related: https://math.stackexchange.com/questions/6870/in-written-mathematics-is-fxa-function-or-a-number/6876 — Gaurang Tandon, Feb 12 '18 at 05:32
In any case you would need at least a triplet for general functions, (Domain, Codomain, Rule), wouldn't you? In any case, how about thinking about functions as subsets of a cartesian product? — LaurentP, Feb 12 '18 at 05:40
Yes you can. And if you allow some numerical imprecision you can even bridge to $\mathbb [a,b] \to \mathbb [a,b]$ for subsets of $\mathbb R$ by interpolation. (of course assumes a well behaved enough function for the resolution ) — mathreadler, Feb 12 '18 at 05:55
What do you mean by "rule to map one integer to another integer"? To me this sounds exactly like a function, so you could think of $f: \mathbb{Z} \rightarrow \mathbb{Z}$ as the singlet ("rule to map one integer another integer"). Maybe you are interested in the concept of relations. (https://en.wikipedia.org/wiki/Binary_relation) — Peter, Feb 12 '18 at 07:08

Q the Platypus · Answer 1 · 2018-02-13T00:17:59.490

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 $\langle \mathbb{Z} \times\mathbb{Z}, x \mapsto x^2 \rangle$

edited Feb 13 '18 at 00:17

answered Feb 12 '18 at 05:39

Q the Platypus

4,626

It should be a 3-tuple, otherwise how would you distinguish a function $Z^2\to Z$ from a function $Z\to Z^2$? – Feb 12 '18 at 05:50
I am not sure I understand this answer. Could you elaborate your answer? For example, which 2-tuple would I identify with the function $f:n \mapsto n^{2}$? – Peter Feb 12 '18 at 07:11
(Z x Z) x Z vs Z x ( Z x Z) the cross produc doesn’t commute. – Q the Platypus Feb 12 '18 at 12:07
@Peter Added an example to improve clarification. – Q the Platypus Feb 13 '18 at 00:18

Peter · Answer 2 · 2018-02-12T09:42:08.763

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.

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

2 Answers2