Suppose we have a set of sets
$$S = \{S_1, S_2, ..., S_n \}$$
and for ease of notation, let
$$\Omega = \bigcup_{i=1}^n{S_i}$$
that is, the set of all elements that occur in any $S_i$.
And we want to find
$$T_k = \{x \in \Omega \mid x \text{ occurs in at least $k$ of the $S_i$} \}$$
For example, if we had:
$$ \begin{align} S_1 &= \{1, 2, 3\} \\ S_2 &= \{2, 3, 4\} \\ S_3 &= \{1, 5\} \\ S_4 &= \{1, 2\} \\ S_5 &= \{4, 5\} \end{align} $$
then
$$T_3 = \{1, 2\}$$
For the special cases of $k=1$ and $k=n$, we have
$$ T_1 = \bigcup_{i=1}^n{S_i} \\ T_n = \bigcap_{i=1}^n{S_i} $$
so it's kind of a stricter version of a union or a looser version of an intersection.
My question is:
Is there a mathematical term for this operation/function?
A kind of auxiliary question:
I realise you could break this process down a bit by defining
$$C(x) = |\{S_i \mid x \in S_i\}|$$
as a sort of "counter" function.
Then we can re-express $T_k$ in terms of this
$$T_k = \{x \mid C(x) \ge k\}$$
Is there a mathematical term for this function $C(x)$?
I'm not asking for a more concise way to write $T_k$ - this question has arisen in a programming context, so how to do the calculation isn't a concern. What I'm curious about is whether this is a pre-existed mathematical notion and whether it has a name.
I'd be happy to know if there's something defined using $=$ not $\geq$, i.e.
$$T_k = \{x \mid C(x) = k\}$$
so $x$ occurs in exactly $k$ of the $S_i$. The specifics of the comparison aren't so important.
These questions touch on similar ideas but neither really gets close to calculation in my question:
How to write a counter in proper notation?
How to write down formally number of occurrences?
Answers to both of these mention the Iverson bracket, which I'm sure I could use to write $T_k$ more concisely, but as I've said, that is not the question I'm asking.
