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As I will write constraint sets very often, I would like to make sure that I respect the convention.

First, I would like to represent a set of constraints and their relation are conjunction. For instance, $[1 \leq i \leq 2, 1 \leq j \leq 2]$ means the set of all possible pairs of $(i,j)$: $\{(1,1); (1,2); (2,1); (2,2)\}$.

Do you think $[1 \leq i \leq 2, 1 \leq j \leq 2]$ or $\{1 \leq i \leq 2; 1 \leq j \leq 2\}$ is better?

Let's denote the above constraint set (a). Second, I want to say another constraint set "forall i, j satisfying (a), $x \leq i$ and $x \leq j$", which is actually $(-\infty, 1]$ for $x$.

Do you think $\forall i, j \in (a), [x \leq i, x \leq j]$ is the best way to formalize that?

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    In general your sets are written like that: $\{(i,j) : 1 \leq i \leq 2, 1 \leq j \leq 2 \}$, that is you describe the general element of your set then you give the constraints. More info: https://en.wikipedia.org/wiki/Set_builder_notation2012-02-29
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    If you are talking about pairs $(i,j)$, then also use that notation in the description of the set: $A = \{(i,j) : 1 \leq i,j \leq 2\} = \{(1,1); (1,2); (2,1); (2,2)\}$. Then for the set of $x$-values you could write something like $X = \{x : \forall (i,j) \in A: x \leq i,j\} = (-\infty, 1]$.2012-02-29
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    Note that you can also save time and space by writing $\{(1,1); (1,2); (2,1); (2,2)\} = \{1,2\}^2$ which is also accepted notation.2012-02-29

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