# Consider the subsets that contain a specific element

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 $10$.

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Note that the last set in $A$ has only two elements. $\{3,3,10\}=\{3,10\}$. Also note that you are talking about elements of $A$ containing $x$ rather than its subsets. – Asaf Karagila Oct 28 '12 at 16:18
Good point, thanks. How can I refer to the subsets then? Can I say something like: Each $A_i$ element of set $A$ is a subset of cardinality $K$? – limp Oct 28 '12 at 16:47
When you say it is a subset you may want to specify: a subset of what? Every set is surely a subset of itself. As for the cardinality, you can simply say that If $B\in A$ then $|B|=K$ or If $B\in A$ then $B$ is a subset of ... with exactly $K$ elements. – Asaf Karagila Oct 28 '12 at 16:49
Thanks. What if $B$ is a subset of 3 sets? What would be the correct way to express that? For instance, if $C_1 = \{5,6\}$, $C_2 = \{10,12\}$ and $C_3 = \{8,4\}$ then each $B$ has one element of each of the $C_1$, $C_2$, $C_3$ sets. Thanks – limp Oct 28 '12 at 17:03
I've lost you completely. Mostly because of the unclear use of "subset". We say that $B$ is a subset of $C$ if and only if every element of $B$ is an element of $C$. If $B$ has only one element from $C_1$, then it is not a subset of $C_1$. – Asaf Karagila Oct 28 '12 at 17:09

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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@Henning: Thanks! – Asaf Karagila Oct 28 '12 at 16:34
Thanks! I guess that your way of expressing it is a but more generic than $S = \{Y \in A; x \in Y\}$ but both of them seem correct. – limp Oct 28 '12 at 16:50
@limp: The usage of a semicolon is nonstandard. Namely $\{Y\in A; x\in Y\}$ is not a common way to write a set; whereas $\{Y\in A:x\in Y\}$ is, as well the use of a pipe as in my post above. – Asaf Karagila Oct 28 '12 at 16:51
Got it now, thanks. – limp Oct 28 '12 at 16:53