Cartesian product of (possible infinite) family of sets

[Damidami]

Hi all. I'm having trouble understanding the cartesian product of a (possible infinite) family of sets.

Lets say $\mathcal{F} = \{A_i\}_{i \in I}$ is a family of sets.

According to wikipedia, the cartesian product of this family is the set

$\prod_{i \in I} A_i = \{ f : I \to \bigcup_{i \in I} A_i, f(i) \in A_i \}$

My question begins about what information is win/lost within the cartesian product. It seems to me that I can recover the family of sets from the cartesian product (the index set I is there, and for a fixed $i \in I$ I can deduce the set $A_i$ by applying $f(i)$ with every function f in the cartesian product.

If I can construct a cartesian product from the family, and construct the family from the cartesian product, what exactly did I win/loose with constructing it in the first place? Why don't we define the cartesian product simply as the family of sets $\{A_i\}_{i \in I}$?

To clarify my point of view, in the case of the classical cartesian product of two sets A and B, why don't we define the cartesian product $A \times B$ simply as the indexed family $\{ C_k \}_{1 \leq k \leq 2}$ with $C_1 = A$ and $C_2 = B$. Then the element $(a,b) \in A \times B$ would simply mean pick $a \in C_1, b \in C_2$

Any help on clarifying that is appreciated.

 

[Stephen Tashi]

why don't we define the cartesian product $A \times B$ simply as the indexed family $\{ C_k \}_{1 \leq k \leq 2}$ with $C_1 = A$ and $C_2 = B$.

How are you going to define "indexed family"? To "index" something we define a mapping from the index set to the set that is indexed. How will you define the set that is indexed in this case?

Then the element $(a,b) \in A \times B$ would simply mean pick $a \in C_1, b \in C_2$

Doesn't "pick" imply that there exists some sort of function? How are you going to define "picking"?

 

[Damidami]

Hi Stephen,

Thanks for your response.

Usually an indexed family of elements of $X$ is just a function $I \to X$.
In my example I would take $X = C_1 \cup C_2$.

And let's change the word "pick" by the word "choose". So I mean that if you choose $(a,b) \in A \times B$, that means you have chosen $a \in C_1$ and $b \in C_2$.

So I still don't see the need of a cartesian product.

 

[Stephen Tashi]

And let's change the word "pick" by the word "choose". So I mean that if you choose $(a,b) \in A \times B$, that means you have chosen $a \in C_1$ and $b \in C_2$.

Whatever word you like ("pick", "choose", "select") you are asserting that information exists sufficient to perform that process. The standard way to make this assertion is to say that a certain type of function exists.

 

[Damidami]

Let's change the last statement to this one $(a,b) \in A \times B$ would simply mean a function $\phi : \{1,2\} \to C_1 \cup C_2$ such that $\phi(1) \in C_1$ and $\phi(2) \in C_2$.

Wait, that is the definition of the cartesian product, isn't it? just all the functions like $\phi$?

It that is ok, then I can see the need of the cartesian product of a family of sets, it basically allows one to choose many elements of many sets at once, isn't it? (modulo the axiom of choice, that is that the cartesian product of nonempty sets is nonempty)

What I was really trying to do was to define the cartesian product of a family of a family of elements (the principal difference between a family of elements and a set is that a family can *repeat* elements, so to say so). I would like the result (of the cartesian product) to be again a family of elements (instead of a set).

For example, if I have the family $F_1 = \{(1,b) , (2,c), (3,a), (4,b) \}$ and the family $F_2 = \{(1,c),(5,b)\}$, then the cartesian product could be defined be

$\begin{eqnarray*}F_1 \times F_2 &=& \{((1,b),(1,c)), ((2,c),(1,c), ((3,a),(1,c)), ((4,b),(1,c)), \\ && ((1,b),(5,b)), ((2,c),(5,b), ((3,a),(5,b)), ((4,b),(5,b)) \}\end{eqnarray*}$

But that isn't even a function as it assigns two elements to $(1,b)$.

I still don't see how to do it, but I think it can be done. Any help on how to do it? Has it been done? Is it imposible? Or I'm not beeing clear on what I want?

Edit: I would like to add that I think that if I have two families $F_1 : I \to X$, $F_2 : J \to Y$, then the index set of the cartesian product of the two families (to be defined) should be a family whose index set is $I \times J$ (that is the cartesian product of the two index sets). But I'm not sure how to define the mapping so that it makes enough sense to be called the cartesian product of the families. I hope it clarifies what I want, and don't make it darker.

Edit2: I know that if $F_1 = \{A_i\}_{i \in I}$ and $F_2 = \{B_j\}_{j \in J}$ I could assign to each $(i,j) \in I \times J$ the element $(A_i, B_j)$, but that doesn't seem to fit what I would like to call the cartesian product of a family. Because in the cartesian product of sets, of each *index* I have many options, in the example, for $1 \in \{1,2\}$ I could assing any element in $C_1$ to be $\phi(1)$. I can't see clearly what is happening and how to fix it, if that makes sense.

Edit3: I would like to add the motivation behind my wish to define the cartesian product of families of elements. As I see it, the families of elements has enough versatility to do (allmost, at least) the things that can be done with sets. I can see if an element is in the family, I can intersect two families to get a new family, and union two families to get a new family (using disjoint union so that it still is a function).

I'm searching if I can expand it to do anything I can do with sets with families instead, so that I can the drop the concept of set (use it only to define family, kind of *encapsulate* the concept of set behind that one of family) and from there on work only with families. I want to do it in a way that if I change any set X by the family identity function $X \to X$, then all the definitions on families (union, intersection, cartesian product, etc) reduce to the classical definitions on sets. I don't know how feasible it is, but is it's not feasible, I want to know how far can it goes, in replacing sets with indexed families, and where does it fall short and I have to stick with sets instead.

As families they are indexed it is easier to define arbitrary unions, intersections, and any kind of operation like sumatory, using families. I think of it like an array in computer programming, and almost any data structure can be build combining it.

What I still cannot find how to define, is the cartesian product, we have it for sets, so if I want to use only families, I need to define the cartesian product of families, and it isn't as easy as I thought.

 

[Stephen Tashi]

A sophisticated mathematician would look to category theory for an answer. A cartesian product in the "category of sets" is probably an example of an object with certain universal properties. This suggests we ought to figure out what a "morphism" is in the "categories of families" and then construct the appropriate universal object.

The only drawback to this approach is that I am not a sophisticated mathematician. Perhaps if we take this approach, I'll learn category theory.