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.
$\endgroup$ 33 Answers
$\begingroup$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 $x_1,\ldots,x_n$.
$\endgroup$ 3 $\begingroup$According to Wikipedia you don't need the commas:$$\alpha = \max \{ f(x) : x = 1 .. n \}$$Alternatively:$$\alpha = \max \{ f(x) : x \in \mathbb{Z} \land 1 \leq x \leq n \}$$
$\endgroup$ 2 $\begingroup$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).
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