Correct notation for the maximum of an indexed set

Question

Which of the following is the correct notation for the maximum of the set $\{a_{i}x^i\mid i\in\{0,\ldots , n\}\}$?

$$\max\limits_{i\in\{0,\ldots , n\}}(a_{i}x^i) \qquad\text{or}\qquad \max\limits_{i\in\{0,\ldots , n\}}\{a_{i}x^i\}?$$

I know I could write $\max(\{a_{i}x^i\mid i\in\{0,\ldots , n\}\})$, but I would rather use the notation with the indexing set underneath the $\max$.

Answer 1

Short Answer: Either of the suggested notations is "correct", in the sense that both are likely to be understood without confusion. My preference would be to write $$ \max_{i\in\{1,\dotsc,n\}} a_i x^i, $$ with no extra brackets or parentheses.

Discussion: A lot of folk get really hung up on what is "correct" notation. The reality is that there really isn't any such thing, at least not in a any universal sense. The goal of mathematical notation is to clearly and unambiguously communicate an idea. Any notation which does this is "correct".

In the example given above, the goal is to clearly denote the maximum element of a collection of terms of the form $a_i x^i$, where $i$ ranges over a set of integers between $1$ and $n$. Each of the notations suggested in the question seem fine with respect to clearly conveying this idea.

Some additional thoughts:

• My preference would be to write $$ \max_{i \in \{1,\dotsc,n\}} a_i x^i. $$ That is, I think that the notation looks much better without any kind additional brackets. The expression you are trying to maximize is short and unambiguous, so I don't think that there is any difficulty here.

• You could just as easily write $$ \max_{i\in\{1,\dotsc,n\}} \left(a_i x^i \right).$$ In this particular case, I think that the parentheses are a little redundant, but they don't get in the way. On the other hand, if you are maximizing a more complicated expression, they may help. For example, if I write $$ \max_{i\in \{1,\dotsc,n\}} a_i x^i + 1,$$ do I mean "maximize the expression $(a_ix^i + 1)$", or "maximize the expression $a_i x^i$, and then add $1$"? In this case, the notation $$ \max_{i\in\{1,\dotsc,n\}} \left(a_i x^i + 1\right)$$ is an improvement.

• Personally, I don't like the notation $$ \max_{i\in \{1,\dotsc,n\}} \left\{ a_i x^i \right\}. $$ There is nothing really wrong with it, but I find it unaesthetic. If I am going to write $\max A$, where $A$ is some set, I would like to completely specify the set $A$. I don't like the specification of indices under the $\max$ operator, with the set only partially specified after. In this case, I would rather write $$ \max \left\{ a_ix^i : i \in \{1,\dotsc,n\} \right\}. $$ Again, let me be clear: this is entirely a matter of taste. If you like the notation $\max \{a_i x^i\}$, then there is absolutely nothing wrong with it.