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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.)

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    Are you asking about "one-to-one" vs. "one-to-one correspondence"?2011-09-09
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    @Bill, I did not specifically have "one-to-one correspondence" in mind (though now that you mention it, the latter does add to the confusion -- with "correspondence" it is always a bijection, right?). I'm asking for some way to make linguistic sense of "one-to-one" meaning "each element of the codomain is the image of either exactly one _or exactly zero_ elements of the domain".2011-09-09
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    Not that this'll reduce the confusion, but there used to be a tradition that "one-to-one" and "one-one" meant "bijective" and "injective", in some order.2011-09-09
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    @Chris, wonderful. That makes even less sense. I always thought the dash in "one-one" (or "$1 - 1$" as some authors write it) was an abbreviation of "to" in the first place. The term for "surjective" that goes most often with "one-to-one" in my experience is "onto", used as an adjective rather than a preposition.2011-09-09
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    "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.2011-09-09
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    @Arturo, but if I have a function exp: $\mathbb R\to \mathbb R$, then it is _not_ the case that each element of $\mathbb R$ has exactly one preimage; for example $-13$ has zero preimages. Yet the exponential function is injective! True, the _image_ is not the entire codomain here, but in my view this is just an argument that when one specifies the codomain independently of the image, "one-to-one" by rights ought to mean _surjective_ rather than merely injective.2011-09-09
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    I don't know if this is the historical idea, but this is how I think of it. Every function with specified domain and codomain has a natural "range" - the image of the domain. In that sense every function is *basically* surjective. The real-valued function $x \mapsto x^3$ is obviously not surjective if you set the codomain to be $\mathbb{R} \cup \{\text{monkey, apple, } e^{i\pi/7}, \dots\}$, but why would you? From the function's point of view, any points that are not hit are reasonably unnatural to mention. In this sense, it means "one point $\leftrightarrow$ one point **on the image**".2011-09-09
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    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.2011-09-09
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    Off-topic. I checked [this](http://jeff560.tripod.com/o.html)(see also Georges's answer) and found 1 entry for *one-to-one correspondence*, and none for *one-to-one*. Also, strangely, the website claims that *onto* was first used as a preposition in 1940, and as an adjective in 1942. This term isn't as old I imagined it to be. :) (*One-to-one correspondence* has been around for a while though.)2011-09-09
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    @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". (-;2011-09-09
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    @Arturo, so your position is that "one-to-one" was simply mis-adapted when the old function concepts gave way to "a function $A\to B$ is a subset of $A\times B$ that satisfies such-and-such conditions"? It is uncontestable that today "one-to-one" is used by authors who do view a function as something that has (at least contextually) a codomain distinct from the image of the domain, as witnessed by phrasings such as "... it is clear that $f$ is one-to-one, but we have yet to show it is also onto ..."2011-09-09
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    I wondered once why "Sale" on a sign meant that the price was reduced, since the word *really* means "dirty."2011-09-09
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    @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".2011-09-09
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    @Chris has a good point. The terminology is completely natural for those of us, like me, who don’t automatically think of functions as having codomains. To me a function is first and foremost a set of ordered pairs in which distinct pairs have distinct 1st elements; if distinct pairs also have distinct 2nd elements, it’s 1-1, and it’s a 1-1 correspondence between its domain and its range. I still think of a codomain as an extra bit of machinery, and as a result I don’t think of *onto-ness* as a property inherent in the function itself.2011-09-09

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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.

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    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 $f(3)=9\ne\{0,1,4\}$.2011-09-18
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    (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.2011-09-18
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    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 http://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.2011-09-19
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    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.2011-09-19
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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.

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    That's a great answer to "How did "injective" come to mean injective?". But that's not the question being asked.2011-09-09
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    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.2011-09-09
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    @Chris: I have acknowledged what you wrote in my answer.2011-09-09
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    @ Henning: the link in my answer gives the history of "one-to-one": I have edited my text to make this explicit. As to your remark about the history of "injective" being well documented on the net, it is remarkably accurate, but I am not sure what this is supposed to imply. My ambition has never been to give information which can't be found anywhere on the net: this would amount to a permanent ban from this site for me, and presumably for others too.2011-09-09
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    @Georges It would be helpful to clarify if your entire answer is merely an excerpt of Jeff Miller's pages, or whether it contains something beyond that - perhaps independently researched. This is not clear from what you have written.2011-09-09
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    @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.2011-09-09
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    @Georges Thanks for the clarification. That's useful to know, since sometimes the information on JM's pages in not completely accurate.2011-09-09