Increasing numbers of students seem to be using the word "unique" incorrectly. A common example:

Question. Define the term "one-to-one" for a function $f$.

Answer. Every $x$ has a unique $f(x)$.

The answer is nothing new, and in the past it would have indicated that the student has the idea the wrong way around: "every $f(x)$ has a unique $x$" would be fine.

It seems to me, however, that nowadays this answer often indicates not that the student has the wrong idea, but that they think "unique" means "different":

different $x$ values have different $f(x)$ values

is a perfectly reasonable answer to the question. Possibly the error has been "popularised" by web designers who like to proclaim "this site has had 1000000 unique visitors", when they actually mean "1000000 different visitors".

Is there anything (apart from keeping on about it to students, and marking their assessments wrong) that we can do about this? Or do we just have to accept that the English language has changed?

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    $\begingroup$ I'm confused. Doesn't one-to-one mean bijective? Yet the question seems to use it to mean injective. $\endgroup$ Aug 20 '19 at 7:36
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    $\begingroup$ @PeterTaylor: No, one-to-one does not mean bijective. One-to-one and onto does mean bijective. $\endgroup$ Aug 20 '19 at 9:59
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    $\begingroup$ Ah, it's overloaded. A one-to-one correspondence is bijective, but a one-to-one function isn't necessarily. Maybe that's why everyone I know prefers the Latin-derived terms. $\endgroup$ Aug 20 '19 at 10:14
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    $\begingroup$ "Different" isn't even the right word. "Distinct" is the word they want. Another source/reinforcement of your problem in various SQL scripts DISTINCT and UNIQUE are interchangeable. "Different" would be like SELECT A1.COL_A FROM TEST.SOME_TABLE A1 ,TEST.SOME_TABLE A2 WHERE A1.COL_A <> A2.COL_A;, when COL_A is character list from A to Z, you'd get B thru Z, then A and C thru Z...getting the whole alphabet minus the current cycled letter. $\endgroup$ Aug 20 '19 at 17:12
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    $\begingroup$ @JRE You seem to say that students know, as part of their understanding of English rather than via mathematics, what one-to-one means. This would be contrary to what I've seen in beginning students' work and also contrary to what I see in the first (and only) dictionary I checked. On the other hand, I agree with you that there is a language problem here; in my opinion that probem is that the students don't know what "unique" means. $\endgroup$ Aug 21 '19 at 17:24

I don't see this as a major issue, nor do I believe that the word "unique" is in any particular need of saving. There are a large number of terms in mathematics which correspond to vernacular English words, but which have distinct technical meaning in mathematics. For example, an "odd" number is an integer which is not a multiple of $2$, rather than a number which is in some way strange or peculiar (though, oddly enough, the mathematical definition of "odd" predates the modern vernacular usage). As another example, in mathematics "onto" is a synonym of "surjective", while in vernacular English "onto" is a preposition which is of general utility. Or, as a particularly perverse example, what does "normal" mean (hint: even in mathematics, this little word has a lot of different meanings, depending on context)?

Note, also, that this isn't an issue unique (heh heh) to mathematics: in archaeology (and geology, maybe?), the word "flint" refers to a specific type of toolstone which is associate with limestone deposites; in vernacular English, many types of crypto- and microcrystalline silicates are referred to as "flint".

One of the jobs of an educator is to introduce their students to the technical jargon of their field, and to help students to understand that words may have a precise definition which is different from the vernacular meaning, or different from the technical meaning in another field (indeed, "unique visitors" is a well-understood term and has its own technical meaning).

In the example question and answer posed above, I would regard it as an opportunity to discuss this distinction between vernacular usage and mathematical usage. For example, on an exam, I might write something like the following:

Question: Define the term "one-to-one" for a function $f$.

Answer: Every $x$ has a unique $f(x)$.

Response: I understand what you mean by this answer, but this is not the correct usage of "unique" in mathematics. It would be better to say "Every $x$ has a different $f(x)$," or even better to say "Every $x$ is mapped to (or sent to) a distinct value by $f$."

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    $\begingroup$ Actually I am confused (I am not a native speaker). But isnt the original statement ' every x has a unique f(x)' correct? Unique in the sense of 'one of a kind'. $\endgroup$
    – lalala
    Aug 20 '19 at 15:38
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    $\begingroup$ @lalala It is a matter of interpretation, I suppose---the phrase is pretty imprecise. I would say that "each $x$ has a unique $f(x)$" is the definition of a function, i.e. if $f(x) = y$ and $f(x) = y'$, then $y=y'$ (contrast this with a more general relation, where the "image" of $x$ may not be unique). This is why I prefer the more precise language "each $x$ in the domain is mapped to a distinct value in the codomain by the function $f$." $\endgroup$ Aug 20 '19 at 15:48
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    $\begingroup$ @lalala The OED has multiple definitions of "unique". Two are relevant: "Of which there is only one" and "the only one of its kind; having no like or equal" (both dating to the early 1600s). It's the first sense which is primarily used by mathematics (Mathworld:"The property of being the only possible solution"). But it's in the second sense which the student is using it: "for a particular x, that f(x) is the only realization of that value; there is no like/equal among the remaining set of f(x)s". $\endgroup$
    – R.M.
    Aug 20 '19 at 16:33
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    $\begingroup$ @R.M. That is, I think, precisely my point: "unique" has a very specific meaning mathematics, but has many vernacular meanings. The job of the teacher is to distinguish between those, and help students to hone their use of mathematical English. $\endgroup$ Aug 20 '19 at 16:35
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    $\begingroup$ There are a large number of terms in mathematics which correspond to vernacular English words, but which have distinct technical meaning in mathematics. --- More broadly interpreted, this is also the case with mathematical terms, although I've found that many people tend to be unaware of how pervasive this is. For example, not all metrics are non-negative functions. $\endgroup$ Aug 20 '19 at 19:04

To my mind the defintion "Every x has a unique f(x)" of one-to-one is problematic because "has a unique" is neither clear English nor precise. The definition is usually stated as "$f(x) = f(y)$ implies $x = y$", which is unambiguous and avoids any potential interpretative problems related to the use of the word "unique. Alternatively, one could define $f$ to be one-to-one if $x \neq y$ implies $f(x) \neq f(y)$; the issue with this definition is that it only corresponds to the terminology one-to-one after negating both sides.

It is a mistake to try to make mathematical definitions more accesible by framing them in colloquial language. Doing so tends to generate far more confusion than it saves. It is better to advise students (repeatedly if necessary) that mathematical usage and colloquial usage differ, and that part of mathematics involves developing a precise language, and to use this precise language carefully and correctly (even if one is more tolerant of student usage).

  • $\begingroup$ To my mind, the definition "every $x$ has a unique $f(x)$" for one-to-one is not "problematic", it is simply wrong. This was one of the points I was making. $\endgroup$
    – David
    Aug 22 '19 at 0:04
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    $\begingroup$ @David: I think it is problematic rather than clearly wrong. What the student means to say is clear enough and is correct (provided one interprets unique as you indicate, which is common in colloquial use), although the student is unable to use language sufficiently carefully to say what the student means. $\endgroup$
    – Dan Fox
    Aug 22 '19 at 8:13
  • $\begingroup$ Yes, that's exactly what I'm saying in my question. It is clearly wrong if we are using "unique" correctly. It is probably correct in its idea if the student thinks "unique" means different. $\endgroup$
    – David
    Aug 23 '19 at 1:18
  1. I haven't noticed casual everyday use of that word involving a misuse. As for instance, the word "literally" is often misused by millenials.

  2. Even were the word being misused, I wouldn't abandon correct usage. Instead teach and require correct usage. But again, I don't notice the word unique shifting in normal day to day usage.

  3. I think the issue here is more one of fussy correctness in discussing functions. When the kids use the word unique, they mean (tacitly) unique to that x, i.e. single. I would say single (where defined) to differentiate from relationships, like the graph of a circle, that have more than one y to an x. The problem with unique, in the context of a function, is that it implies monotonic increase or decrease (no two or more x's having the same y).

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    $\begingroup$ The meaning of "literally" is shifting (or, rather, it is adopting a second meaning in vernacular English). Language changes. To say that a vernacular usage is a "misuse" is to misunderstand how language changes and evolves. One can be unhappy with that change (for example, the singular "data" sets my teeth on edge), but I don't think it is appropriate to characterize this as a "misuse". In short: "Literally" literally doesn't mean "literally". $\endgroup$ Aug 20 '19 at 15:06
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    $\begingroup$ Duh. I know that it is the beginning of a shift. I said as much. I just wouldn't be so quick to excuse it or to assume that the process is either complete or will inexorably finish. And it sounds like crap to those who know better. For example hiring authorities, reading or speech audiences, etc. So encouraging or being overly protective of such shifts is not beneficial. $\endgroup$
    – guest
    Aug 20 '19 at 15:57
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    $\begingroup$ @guest Well, it's also notable that "literally" has literally been used for multiple centuries in the way you're describing "millenials" using it. The fact that "literally" means both 'truly' and 'falsely but for emphasis' isn't a new thing - use dates back to 1769. $\endgroup$
    – Delioth
    Aug 20 '19 at 18:12
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    $\begingroup$ @guest Would you like me to get offa yer lawn now? $\endgroup$ Aug 20 '19 at 18:59
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    $\begingroup$ Regarding misuse of words and millenials in your answers: XKCD $\endgroup$
    – JAD
    Aug 21 '19 at 14:05

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