I wonder if there's some advantage of using the notation of indexed families when taliking about arbitrary sets or this is just a matter of style. For example to denote $\bigcup A$ and operate over the sets of $A$ some people use $\bigcup\{X\mid X\in A\}=\bigcup_{X\in A}X$, and some others use $\bigcup\{X_i:i\in I\}=\bigcup_{i\in I}X_i$. I'd like to know if they are equivalent in all respect because I don't know if I have to choose one of them or else I have to learn how to use them depending on the situation because probably one notation is better in some cases but in other cases it's not so good.



The notation $\bigcup\mathscr{A}$ is a little different from the notations $\bigcup\{A:A\in\mathscr{A}\}$, $\bigcup_{A\in\mathscr{A}}A$, $\bigcup_{i\in I}A_i$, and $\bigcup\{A_i:i\in I\}$. Of these, the first two are exactly equivalent to each other, and the last two are exactly equivalent to each other; the first two are a special case of the last two. These four explicitly identify a specific function from the index set $\mathscr{A}$ or $I$ to the family of sets being indexed; of course in the first two this is just the identity map that sends $A\in\mathscr{A}$ to itself. The first notation, on the other hand, does not include any such indexing function. In practice it’s often convenient to index the members of a family by a convenient set different from the family; this gives rise to the last two types. There is no harm in introducing such an indexing.

Which form you use is largely a matter of personal preference, though context and the audience for which you’re writing can play a rôle as well.

  • $\begingroup$ I don't think that I have ever really distinguished between $\bigcup_{i\in I}A_i$ and $\bigcup\scr A$ where $\mathscr A=\{A_i\mid i\in I\}$. Mathematically speaking, they are the same thing, too. $\endgroup$ – Asaf Karagila Sep 5 '13 at 22:24
  • $\begingroup$ @Asaf: My first paragraph is about the case in which $\mathscr{A}$ is not already an indexed family. $\endgroup$ – Brian M. Scott Sep 5 '13 at 22:26
  • $\begingroup$ Brian, every family is an indexed family. $\endgroup$ – Asaf Karagila Sep 5 '13 at 22:27
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    $\begingroup$ @Asaf: Not explicitly, no. Of course you can index it by itself, but until you do so it is not an indexed family. $\endgroup$ – Brian M. Scott Sep 5 '13 at 22:28
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    $\begingroup$ That's better... I still feel that it's important to remark that all these notations denote the same set (under the obvious assumption that the sets $\{A_i\mid i\in I\}$ and $\scr A$ are equal, of course). $\endgroup$ – Asaf Karagila Sep 5 '13 at 22:38

Every notation is good, as long as it is understood by the reader.

To know which notation to use and when requires experience. It may depend on the conventions related to the specific topic you are writing about, or to other notation used.

When in doubt, be explicit.


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