# Is it possible to express “any set B composed of any two elements from set A” using set theory notation?

I have a set A with three elements. I'd like to express another set B in terms of a limited number of elements (two in this example) from set A.

In this case I realize that there would be a number of possible B sets, but I'm interested in expressing that B could be any of those sets.

Alternatively, set C which is one of the possible B sets.

Is this possible, or should I be looking for some other notation than set theory?

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Just to be clear, do you want subsets of $A$ with exactly two elements or at most two elements? Also note that you can introduce whatever notation you like just as long as it is consistent. Wether you're inside set theory or not is of little interest when it comes to notation. – Git Gud Feb 5 '13 at 7:33
For this example, exactly two. But I'd be interested to see how you express either. – Gustav Bertram Feb 5 '13 at 7:36
Brian's answer clears both cases up. – Git Gud Feb 5 '13 at 7:37

If $A$ is a set and $n\in\Bbb N$, $[A]^n$ is a common notation for the set of $n$-element subsets of $A$. (In fact this notation is used more generally, with any cardinal $\kappa$, finite or infinite.) If you want the family of all $2$-element subsets of $A$, you can write $[A]^2$. If you want to say that $C$ is a member of that family: $C\in[A]^2$.

This notation is quite standard, but it’s not universally known, so you should probably define it the first time you use it.

Added: In case you find yourself wanting the subsets of $A$ having at most $n$ elements, you can write $[A]^{\le n}$; this is equally standard.

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@Gustav: That’s actually not easy: I’ve known and used it for so long that I can’t give an authoritative reference. I did a little searching, though, and found an example of typical usage in the second paragraph of this PDF. Immediately after Theorem 3.3 of this PDF: ‘As usual, the set of subsets of cardinality $n$ of a set $X$ is denoted by $[X]^n$.’ – Brian M. Scott Feb 5 '13 at 8:03

It's not uncommon to use binomial coefficients for this purpose: $\tbinom{M}{c}=\{S\subseteq M\mid |S|=c\}$. The motivation fur this notation is that $\tbinom{|M|}{c}=\left| \tbinom{M}{c}\right|$. But, as you see in this answer, this notation isn't optimal for inline math. It's, however, quite common in graph theory (see http://mathoverflow.net/questions/36714/notation-for-a-graph-without-any-edges).

Another possibility is writing $\wp_c(M)$ for this set, as it is a subset of the power set.

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