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Given a set $\mathcal{S}=\{1,2,3\}$, I'm interested in the set of all combinations of two elements without repetition:

$\{(1,2),(1,3),(2,3)\}$

Is there a name and symbol for such a set? Something like Cartesian product and $\times$?

I was wondering about $\binom{\mathcal{S}}{2}$, but I guess it is wrong since that is the usual way to define the number of combinations.

I should mention that elements are not necessary numbers.

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One usual notation is $[\mathcal S]^2$.

Generally, $[A]^n$ is the set of all the subsets of $A$ with exactly $n$ elements. One can also write $[A]^{<k}$ for all the sets with less than $k$ elements.

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  • $\begingroup$ And $[A]^{\le k}$ for the collection of subsets of $A$ with at most $k$ elements. And in all of these notations $k$ can be an infinite cardinal as well as a finite one. $\endgroup$ – Brian M. Scott Mar 30 '13 at 22:52
  • $\begingroup$ Thank you! Do you know of a document/website I can reference? $\endgroup$ – Julián Urbano Mar 30 '13 at 22:53
  • $\begingroup$ @caerolus: Jech Set Theory (3rd eds.) pp. 51-52. $\endgroup$ – Asaf Karagila Mar 30 '13 at 22:55
  • $\begingroup$ Lifesaver...Cheers! $\endgroup$ – Julián Urbano Mar 30 '13 at 22:57

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