# Set notation for multiple indices

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 $$A$$.

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.