What's the notation for a set of $n$ minimum values of a function? I know that set notation is $\{\}$, and a minimum can be shown with $\min_{x}f(x)$, but how would I denote the $n$ lowest values of $f(x)$ over a given range?
I guess I'm looking for something like $\{\min^n_xf(x),\forall x\in A\}$, where $n$ is the number of lowest values of $f(x)$, but I'm not sure what the correct notation is.
 A: I presume that this question is primarily related to logic and give a discussion accordingly. Otherwise, it might not be of much help.
A symbolism should not be so heavily laden with representation. So, the proper method is to write a formula $\phi$ as a definite description in each particular case for set abstraction (in lambda abstraction-style, to obtain $\sigma x.\phi(x)$ as a set).
Just to illustrate the point only, we can consider the following example. The formula tells that there are three distinct elements $w, x, y$ and for all $z$ different than $w, x, y$, a function $f$ takes $z$ with respect to the ordering relation '$\,<\,$' to higher values:
$\exists w\exists x\exists y\forall z((w\neq x\wedge x\neq y\wedge y\neq w)\rightarrow ((w\neq z\wedge x\neq z\wedge y\neq z)\rightarrow(f(w)<f(z)\wedge f(x)<f(z)\wedge f(y)<f(z))$
We may convert the formula for $x_{1},\ldots, x_{n}$ in place of $w, x, y$ and use it in set abstraction (denoting multiplicity by '$\,\bar{}\,$'; notice that the following is an abbreviated expression and there is no settled standard on this):
$M_{n}=\{\bar{x_{i}}\mid(\overline{\exists x_{i}}\in S)(\forall z\in S)((i\neq j\rightarrow x_{i}\neq x_{j})\rightarrow (\overline{z\neq x_{i}}\rightarrow \overline{f(x_{i})}<f(z)),\; i, j=1,\ldots, n\}$
It should be remarked that the formula will express a logical definition of the resultant set, not how one comes to build the set.
