What does it mean to "define" something in math?

It's unlikely that the parts of the questions that say:  "Define <something> as"   or "Define <something> to be"  are asking you to do anything.   I know it sounds like they're telling you to do something, but what those phrases really mean is that they have created a name for something and are telling you what that name means.

For example, if I said "Define the function f(x) to equal 1 when x > 0 and 0 when x <=0",  then there's nothing for you to do.   I'm basically telling you how to interpret expressions such as f(3).   When you see f(3), I've just told you that f(x) = 1 when x > 0, therefore f(3) = 1.

If I said define a sequence a(n) so that a(0) = 1 and a(n+1)=a(n)+3,  I've just told you how to interpret things like a(1), a(2), a(3).    So I'm not telling you to actually do anything,  I'm just giving you information so that you can understand the next things that I write.

We don't really do this in English because we don't really need to be constantly coming up with new names and symbols for things.   But if I said to you that from now on when I say the word "ploof",  I mean a "fluffy dog",  then that would be the same as the mathematic directive:  "Define 'ploof' to mean 'fluffy dog'".

Note:  I'm just guessing here.  I could very well be that a question is asking you to come up with your own definitions.   I'd have to see the entire question and surrouding context to be sure.
Just post the full question, it's easier to answer.

Sometimes define just mean the same thing as solve, they want you to solve explicitly for something. Sometimes it is a conceptual question about making some intuition rigorous. Sometimes it means an explicit construction of something, but there are more than one constructions so it's a bit weird to say you "solve" for it.
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Can you post an example of a question that you're stuck on. Not really sure what you're confused about
In math a definition is some expression like A ~ B where A is the "thing you're defining", B is the "mathematical description", and ~ just means that I'm relating these 2 things.

In that case, let's say I have a statement about A (maybe A here is a linear transformation). I'll denote that statement as P(A) which you can see as just a sentence on A like "linear transformations preserve the 0 vector". Then the point of a definition is that for all such statements P, I should have that P(A) is equivalent to P(B). That is, if P(A) is true then so is P(B) and vice versa. Also if P(A) is false then so is P(B) and vice versa.

It operates just like a definition in the dictionary. What can be a struggle sometimes though is defining something. Because when you define something in mathematical context, you want to be sure it satisfies all your mathematical requirements (this is the for all P part).
"Define" isn't asking you to do anything in this context, it's telling you that something corresponds to something else.

If you did ever need to refine something though (to answer your title's question), you simply do it by writing "Define <thing> to mean <other thing>" like they do in the questions you mentioned, ex:

> Define $to be an integer concatenation operator, such that 123$567 = 123567 <the rest of your proof>
They may be asking if you know what "n choose k" means, i.e. n! / (k! (n-k)!), or, less likely, how to write it as a binomial coefficient with "big parens".

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