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How would one denote the set of all subsets of $A$ which have size $2$?

Would $$\binom{A}{2}$$

be a good idea?

stefan
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  • I believe the notation is $[A]^2$. Or atleast, we are using it in our set theory class. – Arkady Feb 24 '12 at 17:21
  • With nonstandard notation (at least I believe this is nonstandard notation), I think that being as explicit as possible as helpful. Thus, I would write $${X \subset A : |X| = 2}.$$ Your notation, however, does emphasize the size of such a set which is a great quality in a notation. – JavaMan Feb 24 '12 at 17:23
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    Dear stefan: It wouldn't be a good idea: it would be a terrific idea! – Pierre-Yves Gaillard Feb 24 '12 at 17:23
  • @JavaMan I am not sure whether your notation is First Order Notation I suspect it is Second Order which may be not feasible in some cases. Asaf Karagila had given nice first order definition along with short hand notation. I think ${x \in 2^{|2|}:|x| =2}$ might be first order notation but I am not completely sure. – Trismegistos Feb 24 '12 at 18:02
  • I made a mistake in my notation I meant ${X\in A^{|2|}:|X|=2}$ – Trismegistos Feb 24 '12 at 18:15

2 Answers2

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$\dbinom A 2$ is standard notation for the set of all size-$2$ subsets of a set $A$, in the usage of combinatorialists.

Michael Hardy
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    It is a great notation. However, heavily used in inline math it tends to either get cramped or to push the lines apart. So while the mathematician in me approves, my inner typographer does not. – Harald Hanche-Olsen Feb 24 '12 at 17:38
  • @Harald: There is still some effect of you use a smaller version such as $\tbinom A 2$ but I find it quite reasonable. – Henry Feb 24 '12 at 18:06
  • @Henry: I disagree. The letter A is too small. Maybe that is because I am getting a bit old? In the same vein, I dislike text-mode fractions except for simple numerical fractions like $\frac56$. – Harald Hanche-Olsen Feb 24 '12 at 18:28
  • I always thought this was the number of unordered pairs from $A$. Which never made real sense in case $A$ was not finite. – Asaf Karagila Feb 24 '12 at 19:26
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    The number of unordered pairs from $A$ is $\dbinom{|A|}{2}$ (where $|A|$ is the number of members of the set $A$). The set of unordered pairs from $A$ is $\dbinom{A}{2}$. – Michael Hardy Feb 25 '12 at 04:43
  • @AsafKaragila I believe $$\binom{\aleph_0}{k} = \binom{|\mathbb{N}|}{k} = \left| \binom{\mathbb{N}}{k} \right| = |{X \subseteq \mathbb{N} : |X| = k}| = \begin{cases} 1 & k = 0 \ \aleph_0 & 0 < k < \aleph_0 \ 2^{\aleph_0} & k = \aleph_0 \ 0 & \aleph_0 < k \end{cases}$$ makes sense, for example. – user76284 Oct 27 '19 at 19:57
  • @HaraldHanche-Olsen : Let's try two different froms: $${}$$ Lorem ipsum dolor sit amet, consectetur adipiscing elit $\binom A n,$ sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. $${}$$ Lorem ipsum dolor sit amet, consectetur adipiscing elit $\dbinom A n,$ sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. $${}$$ – Michael Hardy Oct 27 '19 at 21:58
  • @MichaelHardy I am not sure what you are trying to say with these two examples. As I said over seven(!) years ago, the second (displaystyle) form will really push lines apart, ruining the typography in any but the shortest paragraphs. And I find the first (textstyle) form too small to read comfortably. But as I also indicated then, apart from readability issues I find the notation excellent. Alas, one has to compromise sometimes. – Harald Hanche-Olsen Oct 28 '19 at 06:56
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In set theory it can be often denoted as $[A]^2=\{\{a,b\}:a,b\in A, a\neq b\}$.

Asaf Karagila
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