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Essentially I'd like to know the formal definition of the object $\{A_{i}|i\in I\}$ .This is my context:

1.- From Wikipedia (Here) I understand that a family of elements in $S$ indexed by $I$ and denoted by $\{A_{i}|i\in I\}$ is a function $A:I\longrightarrow S$.

2.- From Hrbacek's book (Introduction to set theory): "We say that $A$ is indexed by $S$ if $A=\{S_{i}|i\in I\}=Ran(S)$, where $S$ is a function on $I$". Here I understand that I have the function $S:I \longrightarrow A$ and $\{S_{i}|i\in I\}$ is defined as the range of $S$.

3.- From Hagen von Eitzen's answer (Here), I understand that a family $\{A_{i}|i\in I \}$ is a function $S:I\longrightarrow A$ and we write $A_{i}$ instead of $S(i)$.

So, I'm very confused because these definitions seems to be different to me, or maybe I'm missing something. I don't know. Could you guys help me clarify this?

Edit: I intuitively understand the concept. My problem basically is that there are some "inconsistencies" that I just don't get in the defintions:

  • In definition $1)$ if $A=\{A_{i}|i\in I\}$ is a function then $\bigcup A=\bigcup \{A_{i}|i\in I\} $. But formally the elements of a function are order pairs and then $\bigcup A$ is a set that doesn't equal the union of all the sets $A_{i}$.

  • In definition $2)$ It's been said that $A$ is indexed by $S$. So here I always thought that a $I$ is the set that index funtions, though here it does make sense to say $\bigcup \{A_{i}|i\in I\}$ because we are talking about the union of all the sets $A_{i}$.

  • When we are talking about sets, the object $\{A_{i}:i\in I\}$ cannot contain repeated elements, but if we talk about a function it does matter the order and repeatition of elements. So, it's confusing.

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up vote 3 down vote accepted

Much like the definitions of "countable", "sequence" and "natural numbers" so does the definition of "family" can be changed from one context to another.

First note that the first and third interpretations are the same, they just use different letters. If we also require that the indexing is injective, that is to say that no set appears twice, then all the three interpretations coincide.

I think that the prevailing use today is generally the first/third interpretation, but you can also find the second one being used quite often. In those cases, this is actually like considering the first/third interpretation with the requirement that the indexing function is injective.

More often then not the writer expects the reader to be able and establish the proper context from the text. Not for every definition, of course, but for the very basic and implicit ones -- like the definition of a family.

To your edit, when we say that $A$ is a family, regardless to how we treat, $\bigcup A$ is the union of the sets in the family, this is sort of an abuse of notation which really eases up on notation and formality once you are aware of it.

This might be a good reason to think of a family a set, rather than a function, too.

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Hey, Asaf! Thanks as always. My confusion here comes because if we're talking for example of a family of sets $A=\{a_{i}|i\in I\}$ then $\bigcup {A}=\bigcup\{a_{i}|i\in I\}$ but if $A$ is a function then formally its alements are ordered pairs and then $\bigcup {A}$ is a weird set that doesn't equal the union of all the indexed sets $a_{i}$. – Daniela Diaz Aug 16 '13 at 11:13
Yes there is a lot of abuse of notation going on. It's not really a big deal. – Asaf Karagila Aug 16 '13 at 14:07
After thinkin about for a while and trying to understand the abuse in notation I have this interpretation , please if you can tell me where I'm wrong: given a set $X$ we say that it is a family of sets indexed by the set $I$ iff there exists a biyective funcion $A:I\longrightarrow X$. In this case $\langle A_{i}:i\in I\rangle=(A)_{i\in I}:=A$ and $\{A_{i}:i\in I\}=\{A\}_{i\in I}:=Ran (A)=X$. – Daniela Diaz Aug 16 '13 at 19:47
Daniela, it seems to me that you are correct about your confusion. The definition of a family may allow repetitions, whereas saying that $A$ is a bijection don't allow that. Generally we can think of a family as a different type of object, which is a function, and when we write $\bigcup A$ for a family type object, $A$, we mean $\bigcup\operatorname{rng}(A)$. – Asaf Karagila Aug 16 '13 at 19:49
Daniela, the point is that when we talk about a family we sometimes want to allow repetitions, so $\{a,a,a,a\}=\{a\}$, but $a_0=a_1=a_2=a_3=a$ is still a family of four sets. If you are confused about union or intersection of "a family" then recall that the repetitions don't change the result. Whenever you see $\bigcup A$ and $A$ is "a family" think of it as the union of its range. That's all. Once you stop getting hanged on that, it becomes obvious and you can move along. If you insist on introducing new notation, terminology and whatnot, it becomes impossible. – Asaf Karagila Aug 16 '13 at 20:06

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