# 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.

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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 $\max(f \restriction [n]).$ – goblin Oct 2 '13 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. – goblin Oct 2 '13 at 2:59

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 $x_1,\ldots,x_n$.