About the definition of the indexed set of a family

I'm a computer science student currently studying universal algebra by reading the book Foundations of Algebraic Specification and Formal Software Development by Donald Sannella and Andrzej Tarlecki.

I'm having a hard time trying to understand the subtleties of indexed families. As far as I know, families are just maps between two sets: $$I$$, the index set; and $$A$$, the indexed set. That means that $$|A|_{i_0}$$, which is the element of $$A$$ indexed by $$i_0$$ could be basically anything, like a number, a set, a collection, etc. And that the particular case where $$I$$ is $$\mathbb{N}$$ it is called a sequence.

However, the definition of the product of an indexed family is:

$$\prod_{i\in I} A_i=\{f:I\to \bigcup_{i\in I} A_i: (\forall i_0\in I)(f(i_0)\in A_{i_0})\}$$

Considering the definition of a family, then the product could be also defined as the set of all families $$(a_i)_{i \mathop \in I}$$ with $$|a|_{i_0} \in |A|_{i_0}$$ for each $$i_0 \in I$$

But then, it is implied that the indexed set ($$A$$) must be a collection (a set of sets), because otherwise, $$|A|_{i_0}$$ could be an element like a number thus the union of $$|A|_i$$ would make no sense, since it only works for sets. So why does the definition of a family (at least the ones I've read on several math textbooks) does not require the indexed set to be a collection rather than just a set?

Also, would this mean that the elements of a sequence must be sets (containing just one or more elements)? I'm use to thinking about elements of a sequence, specially when studying convergence, as just real numbers.

And finally, I will give an example of what I understand so that you could tell me what I'm missing:

I will define a family of countries indexed by their currency name. So:

• $$I$$, the index set would be $$I = \{pound, dollar,euro\}$$

• $$A$$, the indexed set would be $$A = \{\{Spain, Italy, France\},\{UK\},\{US, Canada\}\}$$

• Then, the family would be a mapping defined as: $$(A_i)_{i \in I}= \{pound \rightarrow \{UK\},dollar \rightarrow \{US, Canada\}, euro \rightarrow \{Spain, Italy, France\} \}$$

• Finally, the product of the family $$\prod_{i\in I} A_i$$ will consist on a set of mappings (families) where each one of them has $$I$$ as index set, and maps each index to a set containing just one of the elements of the corresponding subset of $$A$$. So in that way, each of these families will map the indexes to one of the six combinations possibles by picking one element from each subset of $$A$$. Then the product of the family will be a set of exactly 6 families, somewhat similar to the Cartesian product of the subsets of $$A$$.

Families are not maps; unless that book has explicifly given the word family some other meaning, a family of things is simply a set of things. An indexed family of things, on the other hand, is technically a function from the index set to the unindexed set of those same things; the things themselves can be of any type(s). In practice, though, it’s often simpler to think of the indexing simply as a way of attaching labels to the members of the family.

As you’ve observed, the definition that you’ve been given of the product $$\prod_{i\in I}A_i$$ of the indexed family $$\{A_i:i\in I\}$$ applies only to indexed families of sets; that does not mean that you cannot have indexed families of other things, as in your example. However, your set

$$A=\big\{\{\text{Spain},\text{Italy},\text{France}\},\{\text{UK}\},\{\text{US},\text{Canada}\}\big\}$$

is not in itself an indexed family; it’s just a set of sets. It doesn’t become an indexed family until you actually index it. Taking $$I=\{\text{dollar},\text{euro},\text{pound}\}$$, you can index $$A$$ as $$\{C_i:i\in I\}$$, where

$$C_{\text{dollar}}=\{\text{US},\text{Canada}\}\,,$$ $$C_{\text{euro}}=\{\text{Spain},\text{Italy},\text{France}\}\,$$

and $$C_{\text{pound}}=\{\text{UK}\}\,.$$

Then $$\prod_{i\in I}C_i$$ is the set of all functions $$f:I\to\bigcup_{i\in I}C_i$$ such that $$f(i)\in C_i$$ for each $$i\in I$$. There are, as you say, six of them; one is

$$\big\{\langle\text{dollar},\text{US}\rangle,\langle\text{euro},\text{Italy}\rangle,\langle\text{pound},\text{UK}\rangle\big\}\,,$$

and the other five are similar. This is indeed similar to the Cartesian product of the sets $$C_i$$ for $$i\in I$$: for instance, this function corresponds to the ordered triple

$$\langle\text{US},\text{Italy},\text{UK}\rangle$$

in the product $$C_{\text{dollar}}\times C_{\text{euro}}\times C_{\text{pound}}$$.

• such a lucid response! thank sir! Aug 9, 2020 at 3:20
• Thank you for your detailed answer. I thought that family and indexed family were synonyms, but now I see they're not. So indexed families can have as indexed set numbers, collections, etc., but in order to think about the product (and I think union and intersection too) they must be collections, right? Aug 9, 2020 at 16:23
• But in the end you said that that the product of the family is the Cartesian product of each subset of the indexed set ($C_i$). However, I think that they are not the same, since for example one element of it would be (US, Italy, UK), and not {(dollar,US), (euro,Italy), (pound,UK)}. But as the book "Naive Set Theory" states, it's possible to establish a one-to-one correspondence between both products. Aug 9, 2020 at 16:29
• @Alexander: You are of course correct; I’ve no idea what I was thinking when I wrote that last bit. It’s fixed now; thanks for catching it. Aug 9, 2020 at 18:32