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I have a set with unknown cardinality. The cardinality can be from a number a to a number b. How can I indicate the number a (or b)?

The question in texts is: "What is the least value of n(A)" (A is a set with a few possible cardinalities) How can I properly express this in mathematical notation? I used the notation(If you call it a "notation") least of n(A)

I have seen this question: Mathematical notation for the maximum of a set of function values

The things that differ is that I need the notation to express the minimum (or max) number a function can give for a constant x, not the min (max) for a range of x

Also, the set of all possible sets isn't given and is there to be found out by the question solver. (as it implies, each solver may generate a different one)

n(A) is the cardinality of the set A

I do not have a strong mathematical background. so please avoid very confusing notations. thanks in advance

Pooia
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  • The least value in a set $S$ (assuming that one exists) is denoted by $\min S$. So the least value of the set $n(A)$ is $\min n(A)$. Another common notation is $$\min_{a \in A} n(a)$$ which is used especially when $n$ is defined implicitly by some expression, for example, $\min_{x \in [-1,1]} x^2 + 2$. – Paul Sinclair Jun 22 '23 at 11:36
  • In your case of cardinality of $A$ (traditionally denoted by $|A|$), you need to identify the collection $\scr A$ of all sets $A$ over which you are taking the minimum. Then it would be $$\min_{A\in\mathscr A} |A|$$ – Paul Sinclair Jun 22 '23 at 11:44
  • Thanks for your useful answer; but the problem is that the set collection (I didn't know how to type that notation) is to be found out by the question solver. I edited the question to include this – Pooia Jun 24 '23 at 04:30
  • What about simply "Let $\alpha$ be the least possible value of $n(A)$?" There is no need to decorate the notation (here, I'm using $\alpha)$ with all the various things it depends on unless you're using the notation in a situation where the collection of $A$'s can vary within the same discussion. If you're simply dealing the statement of a problem that you're going to give the solution to, I think just picking something like $\alpha$ and saying precisely what it represents is enough. – Dave L. Renfro Jun 24 '23 at 09:22
  • The problem is with RTL languages and my inability to translate mathematical contents into English (as evident from the question I posted) @DaveL.Renfro (I'm going to text the problem). thanks for the time you spent. – Pooia Jun 24 '23 at 11:27
  • Your problem doesn't make sense, now. How can you talk about a minimum, if you do not know what it is a minimum of? If you do not know what sets you are taking the cardinalities of, then there is no minimum cardinality to talk about. – Paul Sinclair Jun 24 '23 at 13:11

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