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I understand that there are functions which map $R^n\to R^m$. However, what theorems, if any exist, are there which describe functions which map $R^{a_1}$, $R^{a_2},\ldots, R^{a_n}\to R^{b_1}$,$R^{b_2},\ldots, R^{bm}$, where $a_j$,$b_k$ are not-necessarily distinct natural numbers? Or more specifically, given some relation $A^i$, let its graph be a subset, $A^{a_i}$, of $R^i$, then what theorems, if any, describe the behavior of functions which map $A^{ac_1}$, $A^{ac_2},\ldots, A^{ac_n}\to A^{bd_1}$, $A^{bd_2},\ldots, A^{bd_m}$ where $c_1$, $c_2,\ldots, c_n$ and $d_1$, $d_2,\ldots, d_m$ are not-necessarily distinct natural numbers? Thanks.

Note that $A^{ac_1}$, $A^{ac_2},\ldots A^{ac_n}$ does not necessarily denote their cartesian product (though it could), but rather a function essentially uses these respective subsets through some general process to map onto $A^{bd_1}$, $A^{bd_2},\ldots, A^{ad_m}$, which again is not necessarily the cartesian product of these subsets.

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Set-theoretically, functions into a family of sets is equivalent to a single function into their cartesian product; and a family of functions from a family a sets is equivalent to a single function from their disjoint union. A map from the family of sets $A^{ac_1}\ldots,A_{ac_n}$ to the family of sets $A^{bd_1},\ldots,A_{bd_m}$ would be equivalent to a single family from the disjoint union $A^{ac_1}\amalg\cdots\amalg A^{ac_n}$ to the cartesian product $A^{bd_1}\times\cdots\times A^{bd_m}$, so no new "theory" is required to handle such families. –  Arturo Magidin Aug 25 '11 at 20:24
@ArturoMagidin, thank you very much, but I am not referring to mapping the disjoint union to the cartesian product, but rather, for example,if a function were designated that would "grab" certain elements of each $A^{ac_i}$ according to some specified relation, and then the function mapped each set of "grabbings" to a point which is an n-tuple, and thus the values of the function for all "grabbings" which had n entries would be a relation in n-space, and thus, it could be hypothetically possible to have multiple relations in different n-spaces. Does this make sense? –  analysisj Aug 25 '11 at 20:37
Note that each $A^{ac_i}$ is a Cartesian product in and of itself, if $c_i$>1. –  analysisj Aug 25 '11 at 20:38
Okay: I officially have no idea what you are talking about. A function is a rule that assigns to every valid input one and only one output. The collection of all valid inputs is the "domain" of the function. The domain can consist of anything you may want; it may even consist of functions. If you want your function to have "grabbings" (whatever that may be) as the domain, then fine, take the set of all "valid grabbings" as your domain. It's still just a function from one set to another set. –  Arturo Magidin Aug 25 '11 at 20:44
There is nothing in the definition of function that tells you that two elements of the domain (or two elements of the image) have to have some sort of "similar type" or anything like that. So it doesn't matter that some of the values in the function are ordered pairs while others are ordered 7-tuples, and still others are 4-tuples; nor does it matter that some of the inputs to the function are numbers, while others are pairs of numbers, and still others are triples. Still, just a rule that assigns to every element of a set one and only one output. Still "just" a regular function. –  Arturo Magidin Aug 25 '11 at 21:17
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