# Origin of modern definition of a function as a graph

In the past, I came across a very elegant direct 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")

• In essence, it goes back to Fourier and Dirichlet, but not with this formalism, see McTutor, The function concept. Even Goursat in 1923 laid it out more informally. For Cartesian products and quantifiers you are looking at something around the time of Bourbaki. Commented Feb 28 at 18:06
• On an unrelated note, I'm kind of intrigued by this "coming across" you mention: isn't this the standard definition in all math curricula? Commented Feb 28 at 21:37
• @Gae.S. I don't know - when I try to find it on the Internet, I usually get different definitions in free books or in university courses/materials. Commented Feb 28 at 22:06
• I'll take your word for it. Commented Feb 29 at 0:48
• Before the formalism of Cartesian products, there was the notion of "relation". You could define a function as a special kind of relation. I do not know whether that was commonly done, however. Commented Feb 29 at 17:35

The precise definition can be found into Bourbaki's Elements of Mathematics: Theory of sets (1968; but the 1st French edition is dated 1939), page 76:

A correspondence between a set $$A$$ and a set $$B$$ is a triple $$\Gamma = (G,A,B)$$ where $$G$$ is a graph (a set of ordered pairs), $$A$$ is the source, and $$B$$ the target of $$\Gamma$$.

Domain and range of $$\Gamma$$ are defined usually.

The basic "ingredients" (as per Michael's answer above) can be found in the early 20th Century textbooks on set theory; see e.g. Felix Hausdorff, Set theory, translation of the 3rd German ed 1937 (1st edition as Grundzüge der Mengenlehre (1914)), page 16:

Ordered pairs make possible the introduction of the concept of function [...] Let $$P$$ a set of ordered pairs $$p=(a,b)$$; for every pair $$p$$ that occurs in $$P$$ ($$p \in P$$), we call $$b$$ an image of $$a$$ and $$a$$ a pre-image (or inverse image) of $$b$$. Let $$A$$ the set of all pre-images $$a$$, $$B$$ the set of all images $$b$$ [...][ we say that we have a mapping from one set onto the other.

In the particular case in which each $$a$$ has only a single image $$b$$, we denote this element $$b$$, which is determined by $$a$$ and depends on $$a$$ by $$b=f(a)$$, and we say that this is a single-valued or univalent function of $$a$$ defined on the set $$A$$.

In conclusion, the definition is 100 years old.

• Nice, I was not aware of this definition by Hausdorff. He's careful to distinguish $f$ from $f(x)$, calls $f$ the "mapping" and keeps the original "function of..." for $f(x)$. Commented Feb 29 at 12:37
• Just looked up the Bourbaki quote and noticed (to my surprise) that what they call graph does not coincide with what is nowadays called graph (as in the OP). Their definition of function is given a bit later in EII.13 Definition 9 (french version you linked to) but coincides in essence with what the OP is asking for. Commented Feb 29 at 13:05

I found the desired definition in the fourth edition of "Formulaire Mathématique" by Giuseppe Peano (1903): F1903, page 126.

This definition is formalized to such an extent that it is even in symbolic form, which can be a problem if you are not familiar with Peanian formalism. Anyway, $$\ni$$ means "such that", $$؛$$ corresponds to $$\times$$ (the Cartesian product), $$;$$ defines a pair, $$\text{Cls}'$$ represents the power set, $$\supset$$ is $$\Rightarrow$$ and with subscripts universal quantification is carried out.

Thus, definition 55.1.0 can be translated into modern formalism in this way: $$a, b \in \text{Set} \Rightarrow u : a \rightarrow b \stackrel{\text{def}}{\equiv} u \in \mathscr{P}(a \times b) \land (\forall x . x \in a \Rightarrow \exists y . \langle x, y \rangle \in u) \land (\forall x, y, z . x \in a \land \langle x, y \rangle \in u \land \langle x, z \rangle \in u \Rightarrow y = z)$$

It is not a totally faithful translation because Peano does not define $$u : a \rightarrow b$$, but the set of functions from $$a$$ to $$b$$, however we use the first notation, so I have only moved the variable $$u$$ from right to left (applying to both sides $$u \in$$).

In the bibliographic notes, Peano says that the issue had already been addressed by Cesare Burali-Forti in an 1899 article: Les proprietés formales des opérations algébriques, Revue de Mathématiques, 6, 1899, p. 141-177.

In the note on page 142, Burali-Forti provides a more complex symbolic definition than Peano's (he also uses projection maps), therefore I will not go into details, but the essence is the same. Furthermore, Burali-Forti attributes the basic idea (function as a set of pairs) to Mario Pieri (I assume in a private communication, where Pieri says that the idea is already present in other authors such as Hermann Schubert, even though the logical, general scope of the idea had not yet been fully understood).

• Nice find! Have you looked into earlier editions of the "Formulario" to see when he first introduced this set theoretic definition? I just had a glimpse at this french edition from 1897. On p. 53 onwards he introduces functions, but not as subsets of the product set. Instead in a more "type theoretic" way, where a function is something that can be applied. He also calls them "function signs", in line with his older use in Arithmetices principia from 1889. Commented Apr 24 at 8:46
• @MichaelBächtold Sure, that's what I did! However, after reading your comment, I went back to double-check just to be thorough, and that's when I found out it was also present in the fourth edition. Additionally, there were some bibliographic notes on the subject. Now I need to edit the response... Commented Apr 25 at 19:06
• This is most likely the definite answer. Thanks! Commented Apr 27 at 12:43

This is definitely not 200 years old. The first definitions of the modern function concept are from Dedekind, Cantor, Peano and Frege from around 1890, and even those definitions are not encoded inside the formalism of set theory as graphs. Beware that the original meaning assigned to "function" (by Leibniz, Bernoulli, Euler etc.) was something else.

See this question and answers for some details on this: Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

I suspect that the definition of a function as a special subset of a product set occurred after 1900 (I'd probably start looking in Hausdorff, Zermelo or von Neumann)