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Let's say I have $A = \{x \mid x < 7\} = \{1, 2, 3, 4, 5, 6 \}$ the power set of $A$ has subsets $\{\{\}\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \ldots, \{1, 2, 3, 4, 5, 6\}\}$.

Here I want to have subsets with cardinal of 3 from power set subsets, like $\{\{1, 2, 3\}, \ldots, \{4, 5, 6\}\}$ or with cardinal of 2, what should I write to get only these subsets?

Also I want to know how to apply a function on them, say I have $f(a, b)$, and it can take 2 or more as inputs, I want to apply my $f(a, b)$ on the subsets of cardinal of 2, then of cardinal of 3 and so on?

for example, I want my function take all subsets of $B={x|x<3}$ with size of two as inputs, like $f(1,2)$ and $f(1,3)$ and $f(2,3)$, the order doesn't matter, and the set always has to be finite, then for example I want the sum of all of $f(1,2)$ and $f(1,3)$ and $f(2,3)$, like, would this work $\sum f( (B¦2) )$?

So I have two questions, please answer them.

SchrodingersCat
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Safofoh
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  • how about: ${x\in \mathcal{P}(A):|x|=3}$? – JMP Nov 06 '15 at 19:11
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    http://math.stackexchange.com/questions/884348/notation-the-set-of-two-element-subsets-of-bbb-n http://math.stackexchange.com/questions/707206/notation-to-refer-to-all-the-n-element-subsets-of-a-set and http://math.stackexchange.com/questions/112935/notation-for-all-subsets-of-size-2 include suggestions to standard notations for $n$-elements subsets of a given set. – Asaf Karagila Nov 07 '15 at 16:12

1 Answers1

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Sometimes the notation $A\choose k$ is used for the set of subsets of $A$ of size $k$, in analogy with the notation $n \choose k$ for the number of subsets of size $k$ of a set of size $n$.

This is similar to how $2^A$ is sometimes used to denote the power set of $A$ (the set of all subsets of $A$), when $2^n$ is the number of subsets of a set of size $n$.

I like the $A\choose k$ notation, but I think you should explain what it means the first time you use it, since readers may be unfamiliar with it. [This goes for any notation you may choose for this notion - as far as I know, there's no widely used notation it.]

For your second question, before choosing notation, it would be helpful to clarify what your function $f$ is really doing. It sounds like $f$ can take a variable number of inputs. Are these inputs always a subset of some set $A$? Is $A$ always finite? If not, can $f$ take infinitely many inputs? Does the order of inputs matter (i.e. is $f(a,b) = f(b,a)$?).

Alex Kruckman
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  • the order isn't important, and my function take finite inputs, I've got how to write the notation, but how can apply my function on all the subsets, and get the result as sum of all subsets applied to f, to be more clear, say my function is max(a,b), it can take many inputs, I want the sum of my function max applied to all subsets, in other words, I want the sum of all max numbers in every set – Safofoh Nov 07 '15 at 16:53
  • @Safofoh Well, it sounds like that's a totally different question. You should ask it as a separate question. – Alex Kruckman Nov 07 '15 at 17:49