Notation of maximum with a limited number of elements

Question

I want to express that the maximum of a set is returned. Each element $m_{i}$ of the set is the measure of a physical value (mass). The number of elements in the set is limited and comprise the range $[1,N]$.

$$ y = \max( m_{i}) \quad \forall i\{i\in \mathbb N^{*} | 1 \leq i \leq N \} \\ y = \max( m_{1}, \dotsc, m_{i}, \dotsc, m_{N}) \quad i\in \mathbb N^{*} $$

My question is, if both notations are equal?

Answer 1

I find both of the suggested notations to be confusing, and would not use either. In both notations, the $\max$ operator is acting on some object which isn't completely specified—much of the specification of the argument of $\max$ happens "out of the scope" of the $\max$ operator.

The key idea is that the $\max$ operator acts on ordered sets. That is, $\max$ is a function which takes an ordered set as input, and gives the maximum element of that set as an output.^[1] For example, if $X \subseteq \mathbb{R}$, then $$ \max(X) $$ denotes the maximum of the set $X$. This suggests that the correct notation is $$ \max(X) = x_{\text{max}}, $$ where $x_{\text{max}}$ is the maximum element of the sets $X$.

In the example in the question, this translates to the following:

For each $i \in \{1,2,\dotsc, N\}$, let $m_i$ denote the $i$-th measured mass. Then the maximum measured mass is given by $$ m_{\text{max}} = \max \big( \{ m_i \mid i \in \{1,2,\dotsc,N\} \} \big). $$

There are common ways of simplifying this notation:

• First and foremost, there is a pair of redundant parentheses, which we generally don't bother with. So we write $$ m_{\text{max}} = \max \{ m_i \mid i \in \{1,2,\dotsc,N\} \} . $$

• The nested curly braces may also be confusing, so it would be reasonable to write $$ m_{\text{max}} = \max \{ m_i \mid i \in \mathbb{N}, 1 \le i \le N \} . $$

• Because indexed sets pop up all over mathematics, when operators act on such sets, it is common to subscript an operator with the indices being considered. That is, we might save some space by writing $$ m_{\text{max}} = \max_{i \in \{1,2,\dotsc,N\} } m_i . $$ Even more tersely, $$ m_{\text{max}} = \max_{1\le i\le N} m_i $$ is appropriate when the nature of the index set is well understood.

• Going the other direction, we may just want to ditch the notion of indexing, and write $$ m_{\text{max}} = \max\{ m_1, m_2, \dotsc, m_N \}. $$ This is basically the same notation given in the question (swapping curly braces for round braces—this is a minor change, and either would likely be well understood), except that the indexing set isn't mentioned because it is implicit in the notation.

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[1] There are actually some technical details which I am sweeping under the rug. We would like to think of $\max$ as a function on the powerset of some ordered space, e.g. $\mathbb{R}$. However, not ever subset of $\mathbb{R}$ has a maximum. For example, open intervals such as $(0,1)$ have no maximum. Hence we either have to regard $\max$ as a partial function on the powerset of some ordered space, or be a little more careful about defining the domain of $\max$ (for example, by restricting it to compact subsets of $\mathbb{R}$). Alternatively, we can ditch the notion of the maximum, and instead consider the supremum, which is a closely related concept.

In any event, these are technical details which really don't change the answer above. The main point is that $\max$ is a function which takes sets as input.

Answer 2

The second one is less confusing, but the following is even simpler: $$y=\max_{1\le i \le N} m_i$$ Or: $$y=\max_{i \in\{1,\dots,N\}}m_i$$