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Consider the following set $A$ that contains $N$ subsets of cardinality $K$. In this example, N=4 and K=3:

$ A=\{\{5, 10, 8\}, \{1, 10, 4\}, \{6, 12, 2\}, \{3, 3, 10\}\} $

I would like to write an expression that considers only the subsets of set $A$ which contain a specific value $x$. For instance, if $x=10$, then the subsets $A_1$, $A_2$ and $A_4$ should be considered because they have at least one element that has a value equal to $10$.

Thanks in advance.

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    No need. You're welcome!2012-10-28

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There is no particular notation for this sort of thing. In particular because $A$ itself is not a full power set, but rather a small collection.

In the broader context, one can talk about "all subsets of $A$ containing a point $a$", this is also known as the principal ultrafilter concentrating on $\{a\}$, and many ad-hoc notations are $\mathcal F_a$, or so. In this aspect you could write $\mathcal F_a\cap A$, perhaps.

My usual advice is either to avoid excessive notation, it is best to be abundant but clear, and just give a particular name. E.g.:

Let $S$ be the set $\{B\in A\mid 10\in B\}$.

Or if you prefer to make this a general construction you could perhaps do this:

For a collection of sets $A$, let $A_x$ be the set $\{B\in A\mid x\in B\}$.

This way we don't limit ourselves to a particular $A$ or a particular value of $x$.

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    Got it now, thanks.2012-10-28