Select some particular elements to a set

Question

What I am asking is not a math "problem" but something about presenting the problem in math language.

Assume $0<a_{i}<1, i=1,2,3,...,N$ and $a_{i} \neq a_{j}$, I want to have a set $A$ which contains the $M$ indices with the top $M$ values of $a_{i}$.

For example, we have $a_{1}=0.2, a_{2}=0.6, a_{3}=0.5, a_{4}=0.7$ and $a_{5}=0.1$, and assume $M=3$. Then $I=\{2,3,4\}$.

But how to present $A$ in math language given all $a_{i}$ and $M$?

Thank you for your help.

Answer 1

A natural language description as you posted may well be adequate. A more formal description would be for example:

• Let $I$ be the $M$-element subset of $\{1,\ldots, N\}$ that maximizes $\sum_{i\in I}a_i$.
• Or: Let $\pi$ be the unique permutation of $\{1,\ldots, N\}$ for which $a_{\pi(1)}>a_{\pi(2)}>\ldots >a_{\pi(N)}$. Then let $I=\{\pi(1),\ldots,\pi(M)\}$.

I don't think that any of these ways makes the concept clearer than your natural language decription, so I'd stick with it (or maybe add "in other words, ..." with one of the more foraml descriptions to the definition). If you want to be even more formal, try to get rid of all "$\ldots$" in the defienitions I suggested.