Basic Definition of a Function

Question

In the past, I came across a very elegant definition (below) of a function, which is based on the fundamental concepts of triples, pairs, and sets. However, I find it difficult to search the internet for any citations that I could refer to. One professor once told me that this definition could be over 200 years old, and it might be difficult to determine the author. But I would be satisfied with even citations from reliable sources where this definition was presented or used. Do any of you perhaps recognize any works of this type and could help me?

The function $f= (X,Y,G)$ is an ordered triple consisting of the following elements:

a domain $X$ that is any set
a codomain $Y$ that is also any set
a graph $G \subseteq X\times Y$ being a set of pairs, such that $\forall x \in X, \exists ! y\in Y : (x,y) \in G$

($\exists !$ means "There exists exactly one")

See https://en.wikipedia.org/wiki/History_of_the_function_concept — lhf, Feb 28 '24 at 14:18
Maybe it worth to post this question on https://hsm.stackexchange.com — zkutch, Feb 28 '24 at 14:21
This reads like a pretty typical set theoretic view of a function. Halmos' Naive Set Theory would be a place to start as as reference. This view has been around for about 150 years. — RobinSparrow, Feb 28 '24 at 14:22
See Exact definition of a function and Concept of a function and Idea of a formula as a function. — Mauro ALLEGRANZA, Feb 28 '24 at 14:25
@RobinSparrow Halmos Naive Set Theory in page 30 (is available online) defines function more as a graph W. If we limit the definition of a function only to its graph, there will be a problem with determining whether it is a surjection... — Kamil Kiełczewski, Feb 28 '24 at 14:33
@MauroALLEGRANZA definition in wiki limit function to its graph - this causes a problem with determining whether the function is a surjection or not because the graph does not preserve full information about the codomain. — Kamil Kiełczewski, Feb 28 '24 at 14:36
All I'm saying is that you're describing a bijection between X and a subset of Y, which is a subset of ordered pairs of the Cartesian product of X and Y. The wording is atypical, but the idea is the same. Perhaps I'm overlooking something. — RobinSparrow, Feb 28 '24 at 14:37
Misread your notation I realized. It's not a bijection, but it's a mapping of X into Y, and it's single valued. This seems normal to me. — RobinSparrow, Feb 28 '24 at 14:44
Probably what you want can be found in one or more of the references I gave in this MSE answer. — Dave L. Renfro, Feb 28 '24 at 14:52
You may find it useful to read history of the function concept. — N. F. Taussig, Feb 28 '24 at 23:55

score 1 · Accepted Answer · answered Feb 29 '24 at 03:13

I have seen this formulation quite a few times, though I myself am trying to think about from where altogether. If precisely what you want is a reference for validation, here's at least one located in the footnote of page 2 in the following document.

https://www.heldermann.de/SSPM/SSPM01/Chapter-2.pdf

You are right that this provides an elegant formulation of a function that allows you to unambiguously determine whether a function is surjective or not (as opposed to declaring that the identity function on the rationals is also a function into the reals). However, this doesn't really change how functions are used in practice in any manner.

And that is precisely another advantage of this definition that nothing needs to be changed regarding the use of the function in practice :) — Kamil Kiełczewski, Feb 29 '24 at 09:17

Basic Definition of a Function

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