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Suppose I have a discrete set $A=\{a_1,\ldots,a_k\}$ and I wish to write an expression for the minimum of this set. I would write:

$$ \min \{a_1,\ldots,a_k\} $$

But what if rather than just an element, I want the subset of $A$ containing the $n$ smallest elements of $A$? How would I write that?

Alexander Soare
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    $\left{a\in A\mid # {a'\in A\mid a'\le a}\le n\right}$? – David Sheard Jun 18 '22 at 10:56
  • @DavidSheard I haven't seen # used for cardinality before but I'm assuming that's what it means? I read this as "the set of all $a$ such that for each $a$ there at most $n$ elements in $A$ that are not greater than $a$". I think this works if all elements of the set are unique but breaks down if there are ties, like if $A={1, 2, 3, 3}$ and we want $n = 3$. The desired expression would evaluate to ${1, 2, 3}$ but I think yours evaluates to ${1, 2}$ – Alexander Soare Jun 18 '22 at 11:12
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    @AlexanderSoare by definition, a set has no repeated elements. Otherwise you are using a multiset, and need to change both your question and David Sheard's notation from set brackets to however you represent multisets (e.g. ${|1,2,3|}$). – A.M. Jun 18 '22 at 11:13
  • @A.M. got it. I think I should change my question to be for "expressions" instead of sets. Like for example $\min_{x \in \mathbb{Z}} x^2$. But I think your answer covers that anyway so I believe we're done here. – Alexander Soare Jun 18 '22 at 11:16
  • Yes $#$ is used for cardinality, and @A.M. is right I used the property that sets have no repeated elements to formulate my definition – David Sheard Jun 18 '22 at 11:33
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    I would say "WLOG $a_1 < a_2 < \dots < a_k$" (or if you need their current ordering, define say $a_i'$ to be obtained by sorting the $a_i$) and then you can talk about ${a_1, \dotsc, a_n}$, and maybe define a shorthand for this set if you want. I think that's much clearer than any notation that's more technical or formal. – Izaak van Dongen Jun 18 '22 at 12:32

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I don't think there is a standard notation. It is likely to be most clear if you define your own notation in words before using it. Clear notation that springs to mind for me would be $\min_n (A)$ or $\min^{(n)} (A)$, but it needs explanation.

You can define the concept using only existing notation, but it is perhaps less clear than using words:

$$a_1=\min(A) \\ a_{i+1}=\min(A\setminus\{a_1,\dots,a_i\})\ \text{for}\ i\ge 1\\ \text{min}_n(A)=\{a_1,\dots,a_n\}$$

David Sheard's comment suggests another definition.

A.M.
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