I use set theory on an implicit basis as a programmer; however, it's recently become necessary for me to expand into the formal world to explain my intent to decision makers that don't operate at a software engineering level.
If I have a set of data, called $R$, and it represents a set of equally sized tuples. How would I represent the formula of a second set derived from the elemnets of $R$, where some of the values represent 'groups' from the elements of the tuples within $R$?
An example: I have a list of customer cars as a dataset. And I want to represent a set $D$ which is all of the customer vehicles that are of the Dodge Brand. Doing this algorithmically with a high level programming language with integrated query is a snap, but I don't know how to put it into a formula.
Naturally this will be further subdivided to represent the logical path on how a decision is reached, such as the times a particular customer has brought their vehicle in, the invoices would be such a sub set. This example is a bit contrived, because I can't really divulge the full story due to confidentiality concerns.
This also needs to take into consideration that multiple sets might be used to construct a final result, of which some of the data elements might overlap (such as Vehicle IDs being represented as elements of other sets of tuples.)
Any help?
Edit: I've been working on what I think may describe some of the sets and I wondered if I had the notation right, as follows: $$ R = \left \{ r_{1}, r_{2}, ..., r_{n} \right \} $$ $n \overset{\underset{\mathrm{def}}{ }}{=}$ Total Item Count $$ r_{j} = \left ( T_{t}, T_{s}, D_{G}, S, M, V, B \mid \left\{\begin{matrix} T_{t} \in \mathbb{R},\\ T_{s} \in \mathbb{R},\\ D_{G} : DATE\\1 \leq S \leq 10\\M \in \mathbb{Z} \\ V \in \left \{ Active, Inactive, Trail, Drop \right \} \\B \in \mathbb{Z} \end{matrix}\right. \right ) \mid \left \{ \right.1 \leq j \leq n $$
If you have a set of triplets $R:=\{(x_1,y_1,z_1),\ldots, (x_n, y_n, z_n)\}$, then you can denote the subset of triplets with the second coordinate equaling $5$ by $$D:=\{(x,y,z) \in R \mid y=5\}.$$ If you want to further whittle it down to the ones where the first coordinate is between $0$ and $1$, you can write it as a subset of $R$ or as a subset of $D$. $$\{(x,y,z) \in R \mid y=5, 0\le x \le 1\} = \{(x,y,z) \in D \mid 0 \le x \le 1\}.$$
In some texts/papers, I've sometimes seen data represented like $R:= \{\mathbf{x}_1, \ldots, \mathbf{x}_n\}$, with each $\mathbf{x}_i$ representing a $k$-tuple $\mathbf{x}_i=(x_i^{(1)}, \ldots, x_i^{(k)})$, so the notation can get messy sometimes...