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I have a set $S$ that I want to expand to a $|S|$-tuple. How is the notation for that?

Currently I have something like that: $$ T = (f(x) : x \in S) $$ An example: $$ S = (A,B,C)\\ T = (f(A), f(B), f(C)) $$ In Mathematica the Map Function does what I want.

So basically I want to expand a set of elements and add an entry to a tuple for every element of the set. I need the same functionality a summation does, without building the final sum - if that is of any help.

Can I express this the way I did?

UPDATE

As I want the order of the resulting tuple to be preserved, it is necessary to map from a tuple to a tuple - so $S = (A,B,C)$ instead of $S = \{A,B,C\}$.

Also if you're having the same issue, the Wikipedia Set-builder notation is a good place to start with.

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If you have a set $S$ and you want the set of values that you can get by applying $f$ to elements of $S$, you can write $$\{f(s) \mid s\in S\}.$$ This is the set of all values of the form $f(s)$, for some $s$ in $S$. This notation is standard.

If you say up front “We will use the notation $f^*(S)$ to abbreviate the set $\{f(s) \mid s\in S\}$”, nobody will complain.

Your question asks for tuples, which are different, because the elements of a tuple have an order, whereas the elements of a set don't. (The sets $\{1,2,3\}$ and $\{3,1,2\}$ are the same, but the tuples $\langle 1,2,3\rangle$ and $\langle 3,1,2\rangle$ are different.) But I think you are looking for sets and not for tuples.

If you really do want tuples, you can still do the same thing: You can say “When $T$ is a tuple, we will use the notation $f^*(T)$ to mean the tuple obtained by applying $f$ to the components of $T$ individually.”

Or if you don't like my $f^*$ suggestion, feel free to make up whatever seems convenient. Instead of writing $f^*$ you could write $\operatorname{map}[f]$. Notation is flexible. You can use what seems convenient, as long as you explain it clearly.

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  • $\begingroup$ Thank you very much for your very clear answer. For me, the order is important indeed, so basically I think I would have to redeclare the initial set a tuple, so the mapping is clear. Do you think then it is clear that the order should stay preserved? $\endgroup$ – Thomas Fankhauser Aug 26 '14 at 15:03
  • $\begingroup$ I think, the second part of your answer suggests that? $\endgroup$ – Thomas Fankhauser Aug 26 '14 at 15:05
  • $\begingroup$ The part that is missing from your question is how you know that the result from $\{A,B,C\}$ is $\langle f(A), f(B), f(C)\rangle$ rather than $\langle f(C), f(A), f(B)\rangle$. Is there some ordering defined on the set elements? $\endgroup$ – MJD Aug 26 '14 at 15:12
  • $\begingroup$ You're right. Not for the set. If I say $S = (A,B,C)$, does that make it clear, then? $\endgroup$ – Thomas Fankhauser Aug 26 '14 at 15:16
  • $\begingroup$ If you mean that $S$ is a tuple, and not a set, then yes, it's clear. But you shouldn't say that $S$ is a set if you mean for it to be a tuple. $\endgroup$ – MJD Aug 26 '14 at 16:00

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