# 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. 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? 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. Feb 28 at 22:06
• I'll take your word for it. 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. 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 $f(x)$. 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. Feb 29 at 13:05

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)