Can we save the word "unique"?

Question

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?

Answer 1

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 $f$."

Answer 2

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).

Answer 3

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).