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How would I represent a set of tuples $(a_1,a_2,a_3,...)$, where each element $a_i$ is a positive integer in the interval $1\leq a_i \leq A_i$

The only thing I can think of is defining the set of integers $S_i=\{1\leq k\leq A_i\}$, and then just saying the original set of tuples is, $S_1\times S_2\times S_3\times...$ Where $\times$ denotes the Cartesian product, although this seems like a very messy way to represent this set of tuples, so is there any standard notation or simpler notation for representing a set of tuples like this.

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How about $$ \{a\colon\mathbb N\to \mathbb N\mid \forall i\in\mathbb N\colon a(i)\le A_i\}?$$

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Or perhaps define $b(n)=A_n$, then we have $\{a:\mathbb{N}\rightarrow \mathbb{N}|\forall i\in \mathbb{N}, a(i)\le b(i)\}$. –  vadim123 Apr 25 '13 at 18:01
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I'm pretty sure there's no standard here, so make up any notation you like, as long as it's clear.

One important point is that there's no need to make the notation self-explanatory; that is, if you use ordinary English (or other language) sentences to explain what you mean, the notation can be much more compact, and in my opinion, easier to understand.

For example, you could say:

Let $X=\mathbb{N}^k$, and for any $b=(b_1,\ldots,b_k)\in X$, define $$S_b=\{a=(a_1,\ldots,a_k)\in X\mid a_i\leq b_i \text{ for all }i\}.$$

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I would use $$\prod_{i=1}^n [\![1,A_i]\!]$$

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