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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.

  • 3
    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]).$2013-10-02
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    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](http://en.wikipedia.org/wiki/Natural_number#Constructions_based_on_set_theory), then its not an abuse of notation at all.2013-10-02
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    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^+\}.$2018-08-09

2 Answers 2

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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 $x_1,\ldots,x_n$.

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    Thank you ever so much. It's one of those things where you want to be sure before sending the document out the door.2011-05-23
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    I had a similar problem and thank you!2016-06-28
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    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.2018-08-08
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According to Wikipedia you don't need the commas.

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

Alternatively:

$$\alpha = \max \{ f(x) : x \in \mathbb{Z} \land 1\geq x \geq n \}$$

  • 1
    Never seen that notation before in my life. I would probably infer its meaning from the context, but I'd feel befuddled. Almost as befuddled as why this question from 7 years ago needed a new answer, to be honest.2018-08-08