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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?

Sara
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2 Answers2

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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.)

Brian M. Scott
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  • Thanks for the wonderful clarification, Brian. – Sara Mar 27 '13 at 18:29
  • @Jake: You’re welcome. – Brian M. Scott Mar 27 '13 at 19:36
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    @BrianM.Scott Beautiful explanation!!, just one thing: when you write ${x_n:n\in\Bbb N}$ and you think of that as the range of its associated function, you are of course not thinking of that as a set right?, because in that case if $A={a,b,c}$ and $\forall n\in \Bbb N(x_n = A)$ then ${x_n:n\in\Bbb N}={A}$. But for example $\prod{x_n:n\in \Bbb N}\neq\prod{A}$. So, here my confusion is then because we are talking about ${X_n:n\in \Bbb N}$ both as a set and also as function. Does this make sense? – Daniela Diaz Sep 06 '13 at 18:01
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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 07 '13 at 08:24
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    ... $A^{\Bbb N}$). 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 07 '13 at 08:25
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    @BrianM.Scott :Thank you Brian, it's quite interesting!!, though I'm still confused. Please help me. Let's see for example accordingly $\prod{A_i:i\in {1,2,3}}=\prod{A_1,A_2,A_3}=A_1\times A_2 \times A_3=A^3$ if $A_1=A_2=A_3=A$, for one side. But for the other $\prod{A_i:i\in {1,2,3}}=\prod{A_1,A_2,A_3}=\prod {A}=A$ if $A_1=A_2=A_3=A$. Where am I wrong or what am I missing? – Daniela Diaz Sep 07 '13 at 21:33
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    @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 07 '13 at 21:41
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    @BrianM.Scott: ah, Ok. I think I got it. So basically an indexed set is different than a set that is not indexed, right? It's like in an indexed set repetitions are allowed and the elements are treated as different if they have different index, whereas in a "standard" set repetitions simplify to just having one copy of the repeated elements, meaning they are treated as the same object? Something like ${A,A,A}\neq {A_1,A_2,A_3}$ even if $A_1=A_2=A_3=A$? – Daniela Diaz Sep 07 '13 at 21:57
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    @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 07 '13 at 22:04
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    ... $x:\mathscr{A}\to\bigcup\mathscr{A}$ 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 07 '13 at 22:05
  • @BrianM.Scott. Well, then I think in this case the notation ${A_i:i\in I}$ denotes rather a function and not a 'standard' set like for example when talking about a sequence ${A_n:n\in \mathbb{N}}$ where the repetition of elements matters while in the standard notation it doesn't matter. – Daniela Diaz Sep 07 '13 at 22:23
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    @Daniela: Although the notation ${A_i:i\in I}$ implies the existence of a function with domain $I$, it is nevertheless almost always treated as a standard set, as in the displayed formula in the first half of my previous comment. This is what I meant when I said that in this area notation is a bit inconsistent. And in my notation ${A_n:n\in\Bbb N}$ is not a sequence: it is merely the family of distinct sets $A_n$. That is, if $A_k=A_\ell$, they are the same element of ${A_n:n\in\Bbb N}$. The sequence is $\langle A_n:n\in\Bbb N\rangle$; if $k\ne\ell$, $A_k$ and ... – Brian M. Scott Sep 07 '13 at 22:30
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    ... $A_\ell$ are distinct terms of the sequence even if they happen to be the same set. But as I said, they are not distinct elements of the set ${A_n:n\in\Bbb N}$. – Brian M. Scott Sep 07 '13 at 22:30
  • @BrianM.Scott Perfect!! I think everything makes sense now. Thank you so much for your help and patience, Brian, and sorry for my stubbornness. – Daniela Diaz Sep 07 '13 at 22:41
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    @Daniela: You’re welcome. (And don’t worry about the stubbornness: the important thing is that you understand it!) – Brian M. Scott Sep 07 '13 at 22:47
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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.

H. Rittich
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