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?


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.


1 Answer 1


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.

  • $\begingroup$ 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? $\endgroup$ Aug 26, 2014 at 15:03
  • $\begingroup$ I think, the second part of your answer suggests that? $\endgroup$ Aug 26, 2014 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, 2014 at 15:12
  • $\begingroup$ You're right. Not for the set. If I say $S = (A,B,C)$, does that make it clear, then? $\endgroup$ Aug 26, 2014 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, 2014 at 16:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.