# Mathematical notation for the maximum of a set of function values

I have a question about the proper notation of the following (simplified) example:

I want to express that I have a value alpha, which is the maximum of a set of n values. Each value in the set is the result of a function $f(x)$, and the range of $x$ is between $1$ and $n$.

So something like

$$\alpha = \max(\{f(x) : x = 1,\ldots,n\}).$$

Is this a proper notation? If not, how would I properly express this? It's too long ago for me studying this sort of thing to convince myself I'm writing it down right.

• That's fine, but I'd write it a bit simpler. If $f$ has domain $\{1,\cdots,n\}$, you can just write $\max f.$ Otherwise, the following notation works. Define $[n] = \{1,\cdots,n\}.$ Then you can write $$\max_{i \in [n]}f(i).$$ You could also write $\max(f \restriction [n]).$ Commented Oct 2, 2013 at 2:55
• As an abuse of notation, I think its acceptable to write these as $$\max_{i \in n}f(i)$$ and $\max(f \restriction n)$ respectively, as long as you make it VERY clear that if a function is expecting a set and you give it a natural number $n$, what you're really meaning is the set $\{1,\cdots,n\}$. Or even better, if you begin at $0$ and make use of the von Neumann construction of the natural numbers, then its not an abuse of notation at all. Commented Oct 2, 2013 at 2:59
• It's fine. You can also leave out the curved brackets. In a context where it might not be obvious that every $x$ in the def'n of $\alpha$ must be a positive integer, you could write $\alpha=\max \{f(x): n\geq x\in \Bbb Z^+\}.$ Commented Aug 9, 2018 at 4:51

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 $x_1,\ldots,x_n$.

• Thank you ever so much. It's one of those things where you want to be sure before sending the document out the door.
– Bart
Commented May 23, 2011 at 16:59
• I had a similar problem and thank you! Commented Jun 28, 2016 at 16:02
• I prefer $\max\{f(x_1,\ldots,f(x_n)\}$ with curly braces and no parentheses. In this instance, the parentheses don't actually help, and the curly braces remind you that the thing whose maximum is sought is a set rather than a tuple. Commented Aug 8, 2018 at 20:05

According to Wikipedia you don't need the commas: $$\alpha = \max \{ f(x) : x = 1 .. n \}$$ Alternatively: $$\alpha = \max \{ f(x) : x \in \mathbb{Z} \land 1 \leq x \leq n \}$$

• I think the inequity symbols are incorrectly given in the second equation Commented Aug 20, 2019 at 15:04
• The paragraph in Wikipedia where you found the notation $1..n$ also mentions that this is a notation found in some programming languages. I don't think I've ever seen this notation anywhere except in the context of a programming language. It's so unusual as a math notation that there's no TeX or LaTeX symbol for it; you have to roll your own to typeset it well (tex.stackexchange.com/q/304662/69573). Commented Oct 9, 2021 at 1:26

The most concise notation for this is just

$$\max f[n]$$

where $$f[A]$$ is the image of $$A$$ under $$f$$ and $$n = \{m \mid m < n\}$$ is the ordinal definition of numbers (assuming you start at 0 rather than 1).