# Set notation : Composing set of sets

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

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.

• Well, this is the perfect sign that I need to rest. As I completely forgot the basics. Commented Jul 17, 2020 at 20:38

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.