I'm studying some set theory from Halmos' book. He introduces the generalization of cartesian products by means of families. However, I can't understand what is going on. I get the first introduction "The notation…" to "… one-to-one correspondence". What I'm having trouble is with
If $\{X_i\}$ is a family of sets $(i\in I)$, the Cartesian product of the family is, by definition, the set of all families $\{x_i\}$ with $x_i\in X_i$ for each $i$ in $I$.
Could you explain to me the motivation of this definition? I know families are itself functions $f:I\to X$ such that to each $i$ there corresponds a subset of $X$, $x_i$. Instead of this we write them succintly as $\{x_i\}_{i\in I}$ to put emphasis on the range (indexed sets) of the function and the domain (indexing set) in question.
For example, in my case, the family is $f:I\to X$ with $f(i)=A_i$ with ${\rm dom} f=\{0,1,2,3\}$ and ${\rm ran} f =\left\{ {{A_0},{A_1},{A_2},{A_3}} \right\}$.
I'm thinking that we can talk about the cartesian product of sets as a set of tuples. However, I can't understand the definition for families of sets.
I leave the page in question:
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Best Answer
Your intuition is exactly right. If $I$ is a set and you have a collection of sets $X_i$ for each $i \in I$, then the cartesian product is like a tuple. For example, in the case where you have two sets, $X_0$ and $X_1$, your index set is the finite ordinal $2 = \{0,1\}$. $X_0 \times X_1 = \{(a,b) : a \in X_0 \text{ and } b \in X_1\}$. Another way of thinking about this is $X_0 \times X_1$ is the collection of all functions from $2 = \{0,1\} \rightarrow X_0 \cup X_1$ such that $f(0) \in X_0$ and $f(1) \in X_1$. Instead of tuple, the cartesian product here is a correspondence $f$ between the index set $2 = \{0,1\}$ and an element such that $f(i) \in X_i$ for $i \in \{0,1\}$.
Now to generalize, you want the cartesian product to be the set of correspondence between the index set $I$ and elements in $\bigcup_{i \in I} X_i$ such that $f(i) \in X_i$. So formally, the cartesian product $\prod_{i \in I} X_i = \{f : I \rightarrow \bigcup_{i \in I} X_i : f(i) \in X_i\}$. As you can see, this is a generalization of the tuple concept.