This is a bit of a soft question, but what does or should "unique" mean? Does it mean "at most one" or "exactly one"? I have seen different textbooks use different conventions on this matter.
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5Usually unique means exactly one – Ron Abramovich Jul 28 '21 at 15:59
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Context matters. One could say, e.g., "if there is a solution then it is unique" which clearly does not imply existence. – lulu Jul 28 '21 at 16:04
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2@lulu Your statement doesn't imply existence but uniqueness means exactly one. In fact, I would translate your statement as, if there is a solution then there is exactly one. – John Douma Jul 28 '21 at 16:07
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1An adjective applies to a noun. So just to use the word "unique" you must already be in a context that assumes existence (even if it is a small local context as in lulu's example). My two cents. – Ali Jul 28 '21 at 16:07
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@JohnDouma My point was that use of the word "unique" does not imply existence. Indeed, the most typical use of the term would probably be something like "there exists a unique solution" which at least suggests that existence is something which must additionally be specified. – lulu Jul 28 '21 at 16:09
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2This is an English question because, unlike continuous say, unique is not a mathematical term. It is used by mathematicians but doesn't have a rigorous definition special to mathematics. – John Douma Jul 28 '21 at 16:10
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4I agree with John Douma that this isn't really a mathematics question, but rather a question about English. That being said, context matters. You will often see the phrase "there exists a unique solution"---in this case, "unique" does not imply existence, so the adjective means that there is at most one solution, while the phrase "there exists" gives us at least one solution. As another data point, uniqueness of solutions to PDEs is often shown before existence is even considered---in this context, "unique" again means "at most one". – Xander Henderson Jul 28 '21 at 16:24
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I agree with John Douma and Xander Henderson. Regarding differential equations theory (ordinary and partial), I recall quite well that one very often (in textbooks, class, etc.) talks about the two halves of a result about solutions—existence and uniqueness, with existence almost always being the more difficult half. – Dave L. Renfro Jul 28 '21 at 16:29
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3I think this is on-topic as a question about math terminology, not general English. The words "uniqueness" and "existence" are used in a specific way in math. – Jair Taylor Jul 28 '21 at 17:18
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There is no unique answer since it depends on context and convention. As you wrote, "I have seen different textbooks use different conventions on this matter". – Somos Jul 28 '21 at 19:40
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@Somos, it seems you are right, since there are two posted answers. There exists an answer, but there does not exist a unique answer, to what unique means ;-) – Joe Jul 29 '21 at 18:15
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Of the 11 comments above, only two are entirely accurate: @JohnDouma's. I'm surprised at the lack of consensus on the fact that the actual ambiguity of 'unique'—discussed in my Answer's first link below—has nothing to do with meaning 'at most one'. – ryang Nov 08 '24 at 03:57
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Does “unique” mean at most one or exactly one?
Mathematicians regularly use the word “unique” to mean exactly one, including in the sentence “there is a unique solution, if it exists” (conditional uniqueness), which is logically equivalent to “either exactly one solution exists or no solution exists”, which means that at most one solution exists, which can also be phrased as “there can be only one solution”.
Similarly, consider the sentence “there exists a unique solution”. It just means “there is a unique solution”, that is, “there is exactly one solution”. Contrary to the above comments, another verbatim rephrasing of “there exists a unique solution” is “there is at least one, and exactly one, solution”, rather than “there is at least one, and at most one, solution”—even as these two rephrasings turn out to be mathematically equivalent (to each other and to “there is exactly one solution”).
All in all, the word “unique”, in itself, never means at most one. For the sake of clarity, I don't say “a unique” when I can just say “exactly one”.
Other examples, in mathematical writing, of “unique” as meaning exactly one—rather than at most one:
- The uniqueness quantification, denoted by $∃!$ or $∃_{=1},$ is related to the property of being the one and only object satisfying a certain condition, which in mathematics and logic is termed “uniqueness”.
- Given a function $f$ with domain $A$ and codomain $B,$ every element of $A$ is mapped to a unique element in $B.$
ryang
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3Your "common parlance" is just sloppy English. If you must use the word "unique" in that context, then you should say "every column contains a digit that is unique to that column". It's much clearer to say "every column contains a digit that occurs in no other column". Serious misuses of the word "unique" are widespread in computing and engineering, so we should set a good example in Mathematics. – Rob Arthan Jul 07 '22 at 20:06
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"X has a unique solution" means that X has exactly one solution. That doesn't mean that you can't prove the uniqueness of a (hypothetical solution) before considering the existence of the solution (which is what is going on in the question you link to). – Rob Arthan Nov 04 '24 at 23:47
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- Rob's first comment refers to my since-removed distracting orthogonal point that
– ryang Nov 07 '24 at 14:52"every column has a unique digit" implies "Column X has a unique digit", which can, especially in a non-technical setting, conceivably be read as "Column X has a digit that has no duplicate anywhere".2. Rob's second comment (agreeing with my Answer above) refers to the proof in this Question, which I'd given in a deleted comment as an example of a presentation that contributes to the misconception that 'unique' means "at most one". - Rob's first comment refers to my since-removed distracting orthogonal point that
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The difference between "at most one" and "exactly one" is small but important. As it turns out, "exactly one" implies that there is "at most one", but not the other way around. This makes either acceptable as a stand-in for unique, depending on the context.
"at most one x" means that if you have two things, and both of them are x, they must actually be identical. BUT, there is nothing in "at most one" that guarantees there exists an x. Example: "there is at most one prime divisible by 4" is true, even though there is no such prime.
"exactly one x" guarantees that there exists an x and that there is at most one of these x. Thus, it is a stronger statement than "at most one". So, "there is exactly one prime divisible by 4" is false, since there are no such primes.
Importantly, both of these statements capture the idea behind uniqueness. Often, talk about unique objects presupposes that the object in question exists and so "exactly one" works fine.
But, one occasionally gets statements like "unique, if it exists" which correspond to "at most one".
IsAdisplayName
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It depends how you define "unique".
Uniqueness existential quantification is the concept that there exists one and only one solution or element to satisfy some kind of conditions. The example given on Wiki is that there exists exactly one number n in the natural numbers such that n-2=4; namely, that being n = 6. Thus, it generates a singleton, but not the null set.
This is a more constrained definition of uniqueness than the possibility of there being 0 solutions. In linear algebra, matrices and differential equations often have what is treated as three different types of solutions:
- one (1) unique solution
- no solution (0)
- infinitely many
The result is based on how the matrix / differential is constructed and constrained. In the instance in which one has 0 solutions, there could be close approximations, but those are still not direct solutions. The question then becomes whether or not no solutions is a solution or just a statement of fact. By contrast, if the differential isn't constrained enough, then there could be infinitely many solutions, and thus there is no uniqueness in the solution at all.
Limits are also treated in an interesting way in that sometimes the limit simply isn't computable due to asymptotic divergence, etc., and thus we say that the solution is nonexistent or "the limit does not exist" (DNE), meaning that it doesn't converge to a single solution, discrete and finite, but rather goes off into infinity. That is the answer, but do we call it a solution, or a lack thereof?
Further, there's so-named "essentially unique" where objects are unique solutions only due to the fact that all other solutions are identical through equivalence relations. This is the basis for the Fundamental Theorem of Arithmetic.
I'm sure there are more advanced examples in analysis, number theory, combinatorics, and topology that I have no idea what they mean, including things being equivalent up to a certain isomorphism, but I think that these examples are reasonably expressive in the types of uniqueness that can exist in mathematical discourse.
FluffyFlareon
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