Talk:Indexed family

Notation
There seems to be a mistake in the Notation section in the article. It appears to me that it purports to write two notations, a usual and a misleading one, but both notations look the same to me. -- Jitse Niesen 04:14, 15 Jun 2005 (UTC)

Both notations are common, the first uses round parentheses and the second uses curly parentheses {}. Curly parentheses are also used for sets, which can be confusing. Markus Schmaus 09:15, 15 Jun 2005 (UTC)

I see, thanks. I took the liberty of pointing this out explicitly in the article. -- Jitse Niesen 10:25, 15 Jun 2005 (UTC)

You can't use the notion of family to define ordered n-tuple, since an ordered n-tuple (a triple) is used to define family.--Baterista 16:01, 11 November 2005 (UTC)

The concept of multiset comes in handy between family and set. Family &rarr; multiset &rarr; set. The examples on linear dependence shows the difference between multiset and set, rather than between family and set. Bo Jacoby 11:00, 22 February 2006 (UTC)

I've added some clarification to identify which multiset the indexed family $$\{A_j\}_{j\in J}$$ is. I've used math mode, which may be a mistake. I've also used J, rather than I for the indexed set for consistency with the introduction. I feel that, in the best of all worlds, all this stuff would be in a definition of indexed families, rather than spread out like this...InformationSpace 00:14, 16 May 2007 (UTC)

Definition
JA: The lead of the article begins with an informal definition, but not one so casual that it would be refused service at your average fast food restaurant, and so it contains the word "usually" as a sop to diners from computer science who commonly adapt the idea of an indexed set to their notion of a "union datatype". But that's no reason to sell them the store and get out of the business, if you catch my drift. Jon Awbrey 12:02, 7 June 2006 (UTC)


 * I don't like the "lookup table" being right at the top, since it's neither a mathematical term nor a familiar common usage term. I believe it's a computer science term?  -lethe talk [ +] 12:46, 7 June 2006 (UTC)

Isn't an indexed family the range of a function, rather than the function itself!? It seems to me that an indexed family cannot be simultaneously both a function and a (multi)set (or a tuple, depending on your notation) InformationSpace 05:09, 12 July 2007 (UTC)

Ok, here is my guess at a definition for indexed families. Comments welcomed.

An indexed family  $$\{z_x\}_{x\in X}$$ or $$(z_x)_{x\in X}$$ is the "output" of a surjective function $$f:X\rightarrow Z$$. $$X$$ is the index set and $$Z$$ is indexed by $$X$$. For any $$x\in X$$, $$f(x)$$ is denoted $$z_x$$. $$z_x$$ belongs to the key $$x$$. $$\{z_x\}_{x\in X}$$ (or simply $$\{z_x\}$$) is a set if $$f$$ is injective and a multiset otherwise. $$(z_x)_{x\in X}$$ (or $$(z_x)$$) is a tuple.

InformationSpace 05:35, 12 July 2007 (UTC)


 * The section Mathematical statement defines a "family of elements" simply as a function. Kreyszig only talks about a "family of elements" being "given by" a map from a nonempty set called the index set into another nonempty set (Kreyszig: Introductory Functional Analysis with Applications, A1.3, p. 617). Likewise he uses the word into when referring to the special case of a sequence. He cautions, "one must be careful to distinguish a family from the subset of X whose elements are the elements of the family" (i.e. from the range of the function). The article at Wolfram Mathworld|Family cites a "formal definition" according to which a family is a map f : I --> X (Bourbaki: Eléments de Mathématiques. Théorie des Ensembles). Again, no requirement for surjectivity is stated in the definition itself; but their equation (6) seems to assume it.Dependent Variable (talk) 06:17, 12 November 2010 (UTC)

Origin of Indexed Families?
When were indexed families first introduced and by whom? InformationSpace 04:53, 12 July 2007 (UTC)


 * I am (slightly exasperatedly:-) inclined to believe that the answer as regards the term "indexed family" as distinct from "family" is
 * "It was introduced in the year 2006, by Wikipedia".
 * I was going to this talk page in order to find the arguments that were given for the move of this page from Family (mathematics), and the discussion that preceeded the move, but I found none, except the move edit summary "disambiguation & prevalent usage".
 * I may be slightly unfair. Actually, I am now and then employing the word "family" also for e.g. a set of sets of a particular kind; but I also now and then explain that the reason for considering this as a family is that it may be indexed by itself. I think I've never seen the attribute "indexed", except possibly with a concrete reference: "A family indexed over I".
 * However, I may be wrong. Are there other editors who actually met this combination in articles or text books?


 * In any case, the concept (as distinct from the term) is of course much older than that.JoergenB (talk) 16:24, 25 September 2008 (UTC)

"a family of ... indexed by ..." is just a convention, isn't it?
We can see this kind of statement in too many mathematical texts, but don't know its "formal definition", just because it's an everyday English statement, it's a convention in math, isn't it? Moreover, "a collection of ..." and "a family of ..." are all conventions, not the synonym of set, class, or whatever. When most authors use those statement, I think they don't care what their "formal definition" is. We just need to care only when the axiom of choice exists, and use other terms.--Kilva (talk) 09:45, 10 March 2012 (UTC)

Is an "index set" distinct from an "indexed set"?
I noticed that index set and indexed set link to two different articles on Wikipedia. Is this factually correct, or is it an error? Jarble (talk) 14:52, 27 November 2012 (UTC)

Surjectivity
Where does the surjectivity requirement on the function `x` come from? This looks heavily outdated to me -- it's from the same paradigm as the concept that a function is just a set of pairs and thus doesn't know its domain and codomain. Nowadays, the functions $$\left\{1,2\right\} \to \left\{1,2\right\},\ i \mapsto i$$ and $$\left\{1,2\right\} \to \mathbb{Z},\ i \mapsto i$$ are considered different, so I believe the same should be true of the families $$\left(i\right)_{i \in \left\{1,2\right\}} \in \left\{1,2\right\}^{\left\{1,2\right\}}$$ and $$\left(i\right)_{i \in \left\{1,2\right\}} \in \mathbb{Z}^{\left\{1,2\right\}}$$. And that means we cannot require `x` to be surjective. — Preceding unsigned comment added by Darij (talk • contribs) 17:42, 21 September 2018 (UTC)


 * No element in $X$ is allowed to remain un-indexed, i.e., the indexing function is surjective. Purgy (talk) 18:54, 21 September 2018 (UTC)

Edit
Added simple example to illustrate the difference between indexed family and set, relevant to sequence (-1,1,-1,1,...). Some other things in the article could be cleaned up, but didn't do so. Also, while the concept is completely standard (though not well-referenced), the terminology may be less so. I guess almost all/most/many/quite a few/some... mathematicians don't distinguish between families and sets, but just refer to indexed sets. Reader634 (talk) 11:06, 2 July 2021 (UTC)