I have 5 robots and for each new job, one of them finish last. What is the correct notation for getting the maximum of the times of the 5 robots for a job.
is below acceptable?
$MaxTime = \max\{t_1,t_2,\ldots,t_n\} $
Thanks.
[Math] Mathematical notation: max of 5 values
notation
Related Solutions
Your notation looks fine. You could also use the more informal $\alpha = \max(\{f(x_1),\ldots,f(x_n)\})$ or even $\alpha = \max(f(x_1),\ldots,f(x_n))$.
Finally, you could say that $\alpha$ is the maximum (or maximal) value among $f(x_1),\ldots,f(x_n)$, or that $\alpha$ is the maximum (or maximal) value attained by $f$ on the points $x_1,\ldots,x_n$.
A discussion occurred between us in chat, so here I will highlight the end results.
Depending on how exactly you define a tuple, there are a number of options, but the end result is that the operation is essentially the set of all elements appearing in at least one of $t_1$ or $t_2$ placed into a tuple with entries in the same order as they would appear in the underlying set.
Interpreting tuples as ordered sets:
Letting $t_1 = (A,\prec_a)$ and $t_2 = (B,\prec_b)$ and the underlying set be $(X,\prec_x)$ of which both $A$ and $B$ are subsets of $X$, we have:
$$t_1*t_2 = \left((A\cup B), \prec_x\cap (A\cup B)^2\right)$$
keeping in mind that an order on $X$ is itself a set and is a subset of the cartesian product of $X$ with itself. The intersection them makes sense. Alternatively, one could just refer to the resulting order for $t_1*t_2$ as the inherited order, for which various notations exist such as $\prec_x|_{A\cup B}$.
Interpreting tuples as injective functions from $[n]$ to $X$:
Letting $(X,\prec)$ be the underlying ordered set and letting $t_1=f_a$ where $f_a\in X^{\underline{[n_a]}}$ (i.e. letting $f_a$ be an injective function from $[n_a]$ to $X$) and $t_2=f_b\in X^{\underline{[n_b]}}$, we have:
$$t_1*t_2 = g\in (f_a(A)\cup f_b(B))^{[|f_a(A)\cup f_b(B)|]}~\text{satisfying}~\\\forall j,k\in[|f_a(A)\cup f_b(B)|]~\text{one has}~j<k\implies g(j)\prec g(k)$$
Both of these seem like a lot of words to use for something that can be described much more simply with english rather than symbols: $t_1*t_2$ is the unique tuple such that each entry of $t_1$ and $t_2$ appear once and they appear in the same order as the underlying set.
Best Answer
Depending on context, there a number of things you could write (and this list is not exhaustive):
\begin{align} &\max\{t_1,\ldots,t_n\} \quad\text{ or }\quad \sup\big\{t_1,\ldots,t_n\big\}\\ &\max(t_1,\ldots,t_n)\\ &\max\big(\{t_1,\ldots,t_n\}\big)\\ &\max\big((t_1,\ldots,t_n)\big)\\ &\max\big(\langle t_1,\ldots,t_n\rangle\big)\\ &\max_{i = 1}^{n} t_i \quad\text{ or }\quad \max_{i = 1,\ldots,n} t_i\\ &\max\Big((t_i)_{i = 1,\ldots,t_n}\Big)\\ &\sup\big\{t_i \mid 1 \leq i \leq n\big\}.\\ \end{align} If you want to complicate things and $t_i \geq 0$, there's also the $p$-norm: $$\lim_{p \to \infty}\left(\sum_{i = 0}^{n}|t_i|^p\right)^{\frac{1}{p}}.$$
My personal preference is the first or the second from the list, depending on what is more readable in the context.
I hope this helps $\ddot\smile$
Edit:
Please note that the supremum operator $\sup$ is most often used in a context when we don't know or don't care if the maximum is attained by any element it the set, e.g. you cannot write $\max\left\{-\frac{1}{n} \mid n \geq 1\right\}$ because the set does not contain $0$, yet $\sup\left\{-\frac{1}{n} \mid n \geq 1\right\} = 0$. On the other hand, it is certainly not wrong to write $\sup A$ for a finite set $A$ or any other that contains its supremum, and authors often switch between them depending on whether they want to stress or deemphasize that property.