# Consider all the elements of a set

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 to first question. As long as the membership to a set is clearly stated, almost anything can be part of some set. –  DonAntonio Dec 16 '12 at 12:40
Are the elements of $S$ sets of sets, or ordered sets of sets? –  Asaf Karagila Dec 16 '12 at 12:40
If $S$ is a set, then it does not make sense to speak of "the first element of the $S$ set". A set (as that word is used in mathematics) has no inherent order among its elements. –  Henning Makholm Dec 16 '12 at 13:21
@Asag: Sorry but I don't really know what's an ordered set ;) Each triple is a set itself though...@Henning: Thanks for your input...I am not that good with the correct terminologies...if it's not an element, then what's the correct way for referring to it? –  limp Dec 16 '12 at 15:08
The issue is one of ordering. We usually write $\{ \dots \}$ for an unordered set, and $(\dots )$ for an ordered set. The difference is basically illustrated as follows: $\{1,2,3\} = \{2,3,1\} = \{3,2,1\}$ etc, but $(1,2,3) \neq (2,1,3)$. So it doesn't make sense to talk about the "first" element of an unordered set (though there's no reason why you can't take an unordered set and order it, or an ordered set and drop the ordering). We also only write $\dots\in X$ if $X$ is unordered, but if you write $\in$ when $X$ is ordered, I'm sure it will be fine. It seems unimportant here. –  Billy Dec 16 '12 at 15:21

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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That's great, thank you! Now, if I want to take it a bit further and get the maximum value, not from each of the $t$ triples but only from the ones containing a specific element, $B_{3,3}$ for example, what should I do? –  limp Dec 16 '12 at 15:00
Would something like $\max_{t \in S \mid B_{3,3} \in S}X(t)$ make sense? –  limp Dec 16 '12 at 15:15
How about $\displaystyle \max_{B_{3,3}\in t\in S} X(t)$? (This is a little non-standard because of issues of (non)ordering that the commenters on your question are pointing out... but I think, even without knowing whether or not your triples or your set are ordered, it's clear what you mean.) –  Billy Dec 16 '12 at 15:16
Yes, ranging over "$t\in S | B_{3,3} \in t$" (I think this is what you meant) also makes sense. –  Billy Dec 16 '12 at 15:16
Yes, that's what I meant, thanks! –  limp Dec 16 '12 at 15:17
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