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I've got a set $S$ where each of its elements is a seperate set of 3 elements. For example, consider a case where $S$ is the following:

$ S:=\{(B_{1,1}, B_{2,1}, B_{3,1}), (B_{3,1}, B_{2,1}, B_{5,1}), (B_{1,3}, B_{2,5}, B_{3,3}), (B_{7,3}, B_{4,5}, B_{3,3})\} $

Also, each element of $S$ indicates a specific condition in a system, as indicated by the indexes of the $B$ elements, for which a specific property $P$ takes a specific value. Note that each $B_{x,y}$ is not a number but something else that I have defined.

For example, the first element of the $S$ set is the set $(B_{1,1}, B_{2,1}, B_{3,1})$. This particular combination of $B$s is indicating a specific system state for which a property $P$ gets a specific numerical value, e.g. $X_1$ (the index is $1$ because this is the first $S$ element). So, for the first element of the $S$ set, there is a $X_1$, for the second a $X_2$, etc.

My first question is this: Does it make sense to have a set which doesn't contain numbers (i.e. numerical elements)?

Following, assuming that the above makes sense, what I would like to do is to find the right notation to express the following: For each element of the $S$ set, return the maximum $X_n$.

What I have until now is the following (for a an $S$ set of cardinality $c$): $ \max(\forall n, 1 \leq n \leq c, \; P \; for \; S_n) $

Does that make any sense and is there any way to write it any better?

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    Yes, I agree. Thanks, again.2012-12-16

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Does it make sense to have a set which doesn't contain numbers (i.e. numerical elements)?

Sure. Why not? Actually, we use these all the time - sets of matrices, sets of functions... - but you could have sets of sets of pairs of rainbows and sheep if you wanted to. It's a perfectly well-defined concept, and if you have use for it, great.

what I would like to do is to find the right notation to express the following: For each element of the $S$ set, return the maximum $X_n$.

Then allow me to introduce some intermediate notation: let me take some $(B_1, B_2, B_3) \in S$, and write $(B_1, B_2, B_3) = t$ for short. Then let me write $X(t)$ for the $X$-value associated to the triple $t$. Now your maximum value is $\displaystyle \max_{t\in S} X(t)$. (Or, if you prefer, you can number the elements in your set $t_1, t_2, \dots$ and then write $X_5$ instead of $X(t_5)$, or whatever you like.)

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    Although I prefer your version as it looks more readable and compact ;)2012-12-16