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The question says it all. I have once seen the following notation used for $k$-subsets of a set $S$ but I failed to verify that it is commonly used and I was also unable to find any evidence for a different commonly used notation.

$$ [ S ]^{k} := \{ X \mid X \subseteq S \wedge | X | = k \} $$

Is this notation commonly used? Is there a more commonly used notation? Does this notation clash with another commonly used notation? If there is no commonly used notation would this be a sensible choice because it is similar to the Cartesian product $S^{k}$ but I have not seen $[S]$ used in the context of set theory (besides $[n]$ as the set of ordinals up to $n$)?

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  • $\begingroup$ You can use this, or any other notation (that is not otherwise reserved), provided you define it at first use. If I needed a notation I would probably use something like $S_k$ instead, as square brackets suggest equivalence classes. $\endgroup$
    – vadim123
    Feb 16, 2014 at 22:30
  • $\begingroup$ By the fact that you saw $[n]$ as the set of ordinals up to $n$, it hints me that you've been looking at the wrong places. $n$ is the set of ordinals up to $n$. And $[S]^k$ is a very well established notation for the subsets of $S$ of cardinality $k$. $\endgroup$
    – Asaf Karagila
    Feb 16, 2014 at 22:35
  • $\begingroup$ The similarity with equivalence classes is a good point but $S_{k}$ would not be a good choice because many of my sets have already a lower index. Besides that I am really looking for a notation that is already in use and avoid inventing a new one. $\endgroup$ Feb 16, 2014 at 22:36
  • $\begingroup$ @AsafKaragila So the answer is yes, it is a commonly used notation? Strange that I did not come across any usages during the hours of searching. $\endgroup$ Feb 16, 2014 at 22:41
  • $\begingroup$ Daniel, I didn't come across the notation $n++$ for the successor of $n$ even once when I was looking at all those set theory books; strange how millions of people use that to denote the successor. Or is it? It's really a question of where you were looking. $\endgroup$
    – Asaf Karagila
    Feb 16, 2014 at 22:42

2 Answers 2

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The notation $[S]^k$ is standard in set theory, and appears frequently in discussions of the partition calculus, among other places. Some authors use $S^{[k]}$ instead, to avoid conflict or double brackets when the set $S$ is also an interval.

The notation is also used in the context of finite combinatorics, but not so commonly. The most standard expression in that context seems to be $\binom Sk$.

By the way, in finite combinatorics and other contexts, the notation $[n]$ tends to be used to represent the set $\{1,2,\dots,n\}$. Note that, on the other hand, in the context of set theory, $n=\{0,1,\dots,n-1\}$.

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  • $\begingroup$ For completeness I just came across the notation $\mathcal{P}_{k}(S)$ but again I have no clue how common or uncommon it is. $\endgroup$ Feb 16, 2014 at 23:02
  • $\begingroup$ Yes, the standard meaning of that notation is something else, namely, the collection of subsets of $S$ of size strictly less than $k$. It is used mainly for $k$ an infinite (well-ordered) cardinal, and sometimes additional requirements are added, such as $S\cap k$ being itself an ordinal. Some authors use variants of it, such as $\mathcal P_{=k}(S)$ for the subsets of size precisely $k$, and yet some others (particularly in finite combinatorics) use $\mathcal P_k(S)$ for the subsets of size $k$, but I would strongly advice against this use, given its other meaning. $\endgroup$ Feb 16, 2014 at 23:08
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Yes, the notation is perfectly common and acceptable. It is used often in set theory (especially in contexts where one talks about colorings of $k$-subsets).

As my usual tip here goes, if you're unsure of the validity of certain notations, just add the definition before using it.

If $S$ is a set, we shall denote by $[S]^k$ the set of $k$-subsets of $S$, that is, $[S]^k=\{A\subseteq S\mid |A|=k\}$.

Or something like that.

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  • $\begingroup$ I picked Andres Caicedo's answer because it provides more context, but thanks anyway! And you were right, once I started searching for 'cardinality of subsets' different things started coming up. $\endgroup$ Feb 16, 2014 at 22:59

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