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.
Terminology – Does Unique Mean ‘At Most One’ or ‘Exactly One’?
soft-questionterminology
Best Answer
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".