# Notation for: all subsets of size 2

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

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

be a good idea?

• I believe the notation is $[A]^2$. Or atleast, we are using it in our set theory class. – RKD 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
• 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

$\dbinom A 2$ is standard notation for the set of all size-$2$ subsets of a set $A$, in the usage of combinatorialists.
• @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
• 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 4:43
In set theory it can be often denoted as $[A]^2=\{\{a,b\}:a,b\in A, a\neq b\}$.
• Add condition that $a\neq b$. – Thomas Andrews Feb 24 '12 at 17:49