Cartesian products of families in Halmos' book.

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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Solution 1:

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.

Solution 2:

I think this will help other readers that have this same question (Mendelson, Introduction to Topology):

Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$

as an ordered sequence such that $x_i\in X_i$. Given such a point, by setting $x(i)=x_i$ we obtain a function $x$ which associates to each integer $i$,$1\leq i \leq n$, the element $x(i)\in X_i$. Conversely, given a function $x$ which associates to each integer $i$,$1\leq i \leq n$, an element $x(i)\in X_i$ we obtain the point

$$(x(1),\dots,x(n))\in \prod_{i=1}^n X_i$$

It is easily seen that this correspondence between points of $\displaystyle\prod_{i=1}^n X_i$ and functions of the above type is one-one and onto, so that a point of $\displaystyle\prod_{i=1}^n X_i$ may also be defined as a function $x$ which associates to each integer $i$,$1\leq i \leq n$, a point $x(i)\in X_i$. The advantage of this second approach is that it allows us to define the product of an arbitrary family of sets.

DEFINITION Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product of the sets $\{X_\alpha\}_{\alpha \in I}$, written $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$.

Given a point $x\in \prod_{x\in I}X_\alpha$, one may refer to $x(\alpha)$ as the $\alpha$-th coordinate of $x$. However, unless the indexing set has been ordered in some fashion (as is the case with finite products in our earlier discussion), there is no first coordinate, second coordinate, and so on.