Set notation for multiple indices

Question

I have a set $A_{x}$ where $A_{x}$ is an indexed family of set and represent element $a$ from set $A_{x}$ as $a^{i}_{m}$ . Is this the right notation for elements with multiple indexing ?. All the resources that I have gone through represent multiple indices as subscripts in ordered pairs only (eg; $a_{i,m}$).
Also like set $A_{x}$, I have multiple sets say $A_{1}, A_{2},... $. Let the indexing set be $ I = \{i,j,k\}$ and $J = \{m,n,o\}$. But all combinations from $ I and J (ie.\{i,j,k\}\times \{m,n,o\} $ will not be considered as index for a particular set $A_{1}$. So if I want to define the indexing set for a particular set $A_{x}$, can I do it as follows: Let $I$,$J$, $S_{1}$ and $S_{2}$ be sets. Let $x_{1} : I \rightarrow S_{1}$ be one mapping and $x_{2}: J \rightarrow S_{2}$ be another mapping. (Two mapping functions are used to denote different indices set). Is it correct to write as follows: $$x_{1}(x_{2}(a)) = \langle a_{m}^{i} \rangle_{i \epsilon I , m \epsilon J}$$ What I understood from other resources is that, if I write $I \times J$, it means all possible combinations (Cartesian product) of elements from $I$ and $J$. How can I represent only a few combinations of elements only from $I$ and $J$. ie. for example if $A_{1}$ has indices ${(i,m),(j,m),(k,m}$ and $A_{2}$ has indices ${(i,n),(j,n)}$ only will occur? So set $A_{1}$ = $\{a_{i}^{m},a_{j}^{m},a_{k}^{m}\}$ and family of set $A$ is union of all $A_{x}$. Also what I understood is $A_{x}$ is a family of indexed sets and $A_{X}$ is also an subset of another family of sets $A$.

Answer 1

Well, if you have a collection $(A_x)_x$ of sets, the elements of $A_x$ may be indexed by $a_{x,i}$ if the sets $A_x$ are denumerable. I'd avoid superscripts if possible.