Indexed Family of Sets 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?
 A: 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.)
A: There is no universally valid notation in mathematics. There are so many different mathematical concepts that it is just not possible to find a distinct notation for every one. Thus, a particular notation can have different meanings in different contexts. This means, that a notation means whatever the author defines it to be. If an author defines $\{ A_i \}_{i \in I}$ to be the indexed family with elements $A_i$ for $i \in I$, then that is how $\{ A_i \}_{i \in I}$ is supposed to be interpreted. The same can be said for $( A_i )_{i \in I}$. There are, however, certain types of notation that are more commonly used than others, and deviating from standard notation can cause confusion.
We usually distinguish between sets and tuples. A set can contain a specific element only once, while a tuple can contain the same value in multiple locations, i.e.,
$$ \{ 3, 3, 3 \} = \{ 3 \}
   \quad\text{while}\quad
   ( 3, 3, 3 ) \not= (3) \,. $$
Here, I have used curly braces for denoting sets and parenthesis for denoting tuples, which is a widely used convention.
An indexed family is a function $I \to X$, where $I$ and $X$ are appropriate sets. This means, that an indexed family can have the same value in multiple locations, and is often thought of as a generalized tuple (in the sense that a tuple has indices $1, \dots, n$, while indexed families have arbitrary index sets). Therefore, using a similar notation, i.e.,
$$ ( A_i )_{i \in I}\,, $$
would make sense.
Curly brackets are usually used for sets, thus $\{ A_i \}_{i \in I}$ can be interpreted as the set that contains the elements $A_i$ for $i \in I$, and thus, if $A_1 = A_2 = A_3 = 3$
$$ \{ A_i \}_{i \in \{1, 2, 3\}} = \{ 3, 3, 3 \} = \{ 3 \} \,. $$
This equivalence is, however, not true for indexed families, by definition.
As said in the beginning, defining $\{ A_i \}_{i \in I}$ to be the index set containing the elements $A_i$ is not wrong. It can, however, lead to confusion when the reader mistakes this notation as the set containing the elements $A_i$. Hence, would I assume that this is the reason that some books recommend to use parenthesis to denote indexed families.
In the end, it is all about communication. Parenthesis, angle brackets and square brackets have various other meanings too, that could lead to confusion, and using curly brackets to denote indexed families might be the least confusing notation in certain situations. You just need to communicated your use of notation.
