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Most books write a family of sets $A_i$ with index set $I$ as $\{ A_i \}_{i \in I}$. However, I've read other books that have criticized this notation; they insist that one should write $(A_i )_{i \in I}$ for the family of sets $A_i$ indexed by $I$.

Is there a difference between $\{ A_i \}_{i \in I}$ and $(A_i )_{i \in I}$? If so, could you please give a precise definition of each?

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I don’t like either notation: I would write $\{A_i:i\in I\}$ or $\langle A_i:i\in I\rangle$. Technically there is no difference: each implies the existence of a function $i\mapsto A_i$ whose domain is $I$. The difference is one of emphasis: when I write $\{A_i:i\in I\}$, I’m thinking of this simply as a collection of sets, whereas when I write $\langle A_i:i\in I\rangle$, I’m emphasizing the existence of the function whose domain is $I$ and whose range is that collection of sets. I might let $\mathscr{A}=\{A_i:i\in I\}$ and simply talk about the collection $\mathscr{A}$ of sets, without any reference to the specific indexing, but when I write $\langle A_i:i\in I\rangle$, the specific indexing is very much on my mind: $\langle A_i:i\in I\rangle$ is an abbreviation for a function $I\to\mathscr{A}:i\mapsto A_i$.

For a more familiar example of the distinction, compare $\{x_n:n\in\Bbb N\}$ and $\langle x_n:n\in\Bbb N\rangle$, where each $x_n\in\Bbb R$. In each case $x_n$ is just a handier notation for $\varphi(n)$, for some function $\varphi:\Bbb N\to\Bbb R$. However, when I write $\{x_n:n\in\Bbb N\}$ I’m not thinking of that function; I’m thinking of its range, the set of values that it assumes. When I write $\langle x_n:n\in\Bbb N\rangle$, however, I’m thinking of the function: this is a real-valued sequence, i.e., a function from $\Bbb N$ to $\Bbb R$, not just a countable set of real numbers.

(Note: Many people use parentheses for my angle brackets; I prefer the angle brackets for this specific notational purpose, since parentheses already have more than enough meanings.)

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@Daniela: (Sorry to be so slow.) No, any time I use curly braces (i.e., $\{\ldots\}$) I definitely am thinking of it as a set: the range of a function is a set of points in the codomain of the function. When you form $\prod_{n\in\Bbb N}X_n$, you’re actually forming the set of all functions $x:\Bbb N\to\bigcup_{n\in\Bbb N}X_n$ with the property that $x_n\in X_n$ for each $n\in\Bbb N$. In other words, the indices are actually part of the construction. If each $X_n=A$, you get the same thing when you take ${}^{\Bbb N}A$, the set of functions from $\Bbb N$ to $A$ (which you may write ... – Brian M. Scott Sep 7 '13 at 8:24
... $A^{\Bbb N}$). Again, you have to have some set of indices to use as a domain for the functions that are the points of the product. – Brian M. Scott Sep 7 '13 at 8:25
@Daniela: You’re going wrong when you write $\prod\big\{A_i:i\in\{1,2,3\}\big\}=\prod\{A\}$: the index set is an essential part of the product notation. $\prod\{A\}$ is essentially just $\prod_{i\in\{1\}}A_1$, where $A_1=A$. In order to take the product of three copies of $A$, you have to use a $3$-element index set to ‘separate’ them. – Brian M. Scott Sep 7 '13 at 21:41
@Daniela: Now you’re getting into an area in which the use of notation is a bit inconsistent. Despite the different subscripts, if $A_1=A_2=A_3=A$, one normally would say that $$\big\{A_k:k\in\{1,2,3\}\big\}=\{A_1,A_2,A_3\}=\{A\}\;.$$ It’s specifically in products that the indexing becomes critical, because elements of products are functions whose domain is the index set. This is technically true even when you write $\prod\mathscr{A}$ for some family $\mathscr{A}$ of sets: the implied index set is $\mathscr{A}$ itself, and a point in the product is a function ... – Brian M. Scott Sep 7 '13 at 22:04
... $x:\mathscr{A}\to\bigcup\mathscr{A}$ such that $x(A)\in A$ for each $A\in\mathscr{A}$. $\mathscr{A}$ here is an ordinary set, so its members are distinct; if you want to take the product of identical sets, you’re going to need an external index set, so that you can have $A_i=A_j$ for distinct indices $i$ and $j$. – Brian M. Scott Sep 7 '13 at 22:05

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