# How did “one-to-one” come to mean “injective”?

How did a "one-to-one" function come to mean an injective one? I find it so non-intuitive that I often have to backtrack when reading texts that use "one-to-one" because I suddenly discover that I have been internalizing it as "bijective".

If there was were any logic to the terminology, "one-to-one" would mean bijective and injective would be "(zero-or-one)-to-one".

Perhaps I would be able to remember it better if I knew of any way to make "one-to-one"="injective" make some kind of logical sense, however tenuous. Can anyone suggest one, please?

(To clarify, I know (?) that "one-to-one" is older than "injective", but that doesn't in itself explain how the ancients got the idea of using such a strange and illogical term in the first place.)

• "One-to-one" is in opposition to things like "two-to-one" or "three-to-one"; in general, "$n$-to-one" meant functions for which each image had exactly $n$ pre images. (Keep in mind that even th definition of function was a bit fluid until recently; you also had "one-to-$n$" 'functions' sometimes). In that sense, "one-to-one" just means that each point in the image has exactly one preimage, what we would expect. "Injective" was, I believe, an attempt at clearing that up, trying to capture the idea that an injective function "injects"/"embeds" a copy of the domain in the codomain. – Arturo Magidin Sep 9 '11 at 18:10
• I believe the view that functions don't come equipped with codomains used to be more popular than it is now. That could explain the problem. – Chris Eagle Sep 9 '11 at 18:28
• @Henning: Point the first: $-13$ is not an image of exp; don't confuse "codomain" with "image". Point the second: we're talking terminology that was being used even when the very object to which it was being applied to, functions, was not very clearly defined! It's hardly surprising that the terminology is not exactly logical or perhaps not intuitive to our way of thinking about these objects (functions) which is very different from the way they were thought of then. Not every word in the language means what it "ought to mean by rights". Compare "flammable" vs. "inflammable". (-; – Arturo Magidin Sep 9 '11 at 18:30
• I wondered once why "Sale" on a sign meant that the price was reduced, since the word really means "dirty." – André Nicolas Sep 9 '11 at 18:50
• @Henning: I didn't realize I was taking a "position." The terminology now has a precise meaning ("$f$ is one-to-one" if and only if for all $a,b$, $f(a)=f(b)$ implies $a=b$), which has evolved over time to take into account the evolving understand of what "function" is. I thought you were asking where the terminology came from and why it was used; I gave what my understanding of that is. As the notion of function evolved, the terminology didn't. You seem to complain that the terminology is not "logical", but terminology and language are not always "logical", often they just "are". – Arturo Magidin Sep 9 '11 at 19:03

In the old usage, as well as contemporary usage in set theory, one may consider a function without specifying a particular codomain or target set. (The insistence that a function come along with a particular codomain is a comparatively recent innovation, probably arising in Bourbaki.)

That is, if one understands a function merely to be a set of ordered pairs satisfying the function property (that each input is associated to one output), or as a rule associating to every object in a domain an output value, then it is true to say that a function is one-to-one if and only if it is a bijection from its domain to its range. Thus, injective functions really are one-to-one in the sense that you want.

Of course, this one-to-one terminology was long established by the time Bourbaki wanted to insist that functions come along with a specified co-domain, giving the definition of function as a triple consisting of domain, codomain and set of ordered pairs. The fact that in this context the concept of one-to-one doesn't tell the whole story may be part of the reason that they introducted the injective, surjective, bijective terminology.

But meanwhile, a function is one-to-one if and only if it provides a one-to-one correspondence between its domain and its range. This is perfectly logical, and seems to be the explanation that you are seeking. I would think that the one-to-one terminology begins to seem illogical only when one also insists on attaching to the function a target set or codomain that is not the same as its range, which is, after all, a somewhat illogical thing to do.

• Accepted, though I don't agree that it is in any way illogical to consider a function having domain and codomains as separate attributes. It is true that this isn't done in axiomatic set theory, but axiomatic ST is something like the "untyped assembly language" of mathematics; one has to rely on more or less informal typing rules in order to formalize a standard mathematical argument in strict ST. For example, in many developments of ST, $3=\{0,1,2\}$ by definition, but in standard mathematics, if $f(x)=x^2$, then $f(\{0,1,2\}) = \{0,1,4\}$ but $f(3)=9\ne\{0,1,4\}$. – Henning Makholm Sep 18 '11 at 11:46
• (cont'd) ... Also axiomatic ST cannot easily express everyday inclusion-identifications such as $\mathbb N\subseteq \mathbb Z \subseteq \mathbb Q \subseteq \mathbb R \subseteq \mathbb C$ -- at least not without redefining either equality or membership significantly, which would sort of defeat the entire purpose. None of this is, of course, a relevant criticism of ST for the purpose it was developed to serve, but I think it constitutes a good argument that ST is ill-suited as a final arbiter of which concepts and notations should be considered "logical" in mathematics as a whole. – Henning Makholm Sep 18 '11 at 11:54
• Thanks for accepting! My remark about "illogical" was tongue-in-cheek; but I believe your "in any way" remark to be over-stated, since of course it is perfectly reasonable to consider a functional rule without specifying a codomain. My view is broad and encompassing; please see my answer at mathoverflow.net/questions/30397. It seems inaccurate to suggest that set theorists cannot easily account for the distinctions you mention, and historically, of course, these issues were first resolved by set theorists. – JDH Sep 19 '11 at 0:09
• We probably shouldn't diverge into too long a debate here, but do note that I'm speaking about set theory rather than set theorists. The set theorists I've met have been no less able to work and argue in a variety of styles and traditions than anyone else. – Henning Makholm Sep 19 '11 at 0:19

The terms injectif, surjectif and bijectif are lexical creations of Bourbaki. Their first appearance was in Chapters I and II of his Théorie des Ensembles, published in 1954. (Mac Lane and others knew about them and had used them in print slightly before)

Surjective functions were called fonctions sur in contradistinction to general functions, just called fonctions dans. Bourbaki was very attentive to the quality and beauty of the French he used and found it shocking to use the preposition "sur", instead of a genuine adjective, to qualify the name "function". Hence the neologisms.

Here is a link, in English, to this theme (look at the entry "Injection, surjection and bijection").

As to the terminology one-to-one, your actual question !, the reference I give attributes its first use to Zeuthen in 1870 (in French). The first appearance in English dates from 1873. There is also a reference to its use by Bertrand Russell in 1903.

The above doesn't really answer your question on the reasons why mathematicians used the terminology "one-to-one" but has the advantage of giving hard facts. As to these reasons, the suggestions and guesses in the comments look very reasonable and informed and I second them. And, by the way, your statement that one-to-one predates injective is now proved to be absolutely correct.

• That's a great answer to "How did "injective" come to mean injective?". But that's not the question being asked. – Chris Eagle Sep 9 '11 at 18:16
• Indeed it isn't. The history of "injective" is well documented on the net; that of "one-to-one" is, as far as I can tell, completely opaque. – Henning Makholm Sep 9 '11 at 18:26
• @Chris: I have acknowledged what you wrote in my answer. – Georges Elencwajg Sep 9 '11 at 18:55
• @Bill: I suppose Jeff Miller is the author of the pages I link to? Then the parenthetical remark in paragraph 1 and section 4 are excerpted from there. On the other hand the linguistic remarks on prepositions, adjectives, and Bourbaki's attention to stylistic elegance are my recollections of discussions with people who had been members of or close to Bourbaki . I didn't mention that since it might look like name-dropping. – Georges Elencwajg Sep 9 '11 at 20:17
• @Georges Thanks for the clarification. That's useful to know, since sometimes the information on JM's pages in not completely accurate. – Bill Dubuque Sep 9 '11 at 20:23