Set notation : Composing set of sets

Question

Is there a proper notation to compose sets and produce a set of sets? (I am referring to this as compose due to ignorance of a proper manner to call it)

To illustrate what I want, let me suppose that $\otimes$ does the job, so that

\begin{align} \{1\} \otimes \{2\} &\rightarrow \{ \{1\} , \{2\} \}\\ \{1\} \otimes \{1,2\} &\rightarrow \{ \{1\} , \{1,2\} \} \end{align}

Also, how can we write a composition for a finite number of sets? Say that $U_i = \{i\}$ (trivial example) is there something that can make (again using $\otimes$): \begin{equation} \bigotimes_{i=1}^N U_i = \{ \{1\} , \{2\}, \ldots , \{N\} \} \end{equation}

Answer 1

If I understand properly what you ask for... $I$ is a set . You consider it to be finite in your case.

And you have a set $U$ whose elements are indexed by $I$. Let say $U=\{u_i \mid i\in I\}$.

What you call

$$\bigotimes_{i=1}^N U_i = \{ \{1\} , \{2\}, \ldots , \{N\} \}$$

is then just $U$.

It would ease your understanding if you precise the set that contains the indexes $i$ that you mention.

Answer 2

What's wrong with just writing $\{\{1\}, \{2\}\}$? In the case where the "composition" is over an indexed collection of sets $U_i$, this becomes $\{U_i : i \in I\}$. I don't know of any specialized notation, but I don't think this is too cumbersome.