I have two questions about the set-builder notation for sets of sets. Technically, I believe they are multisets as multiple instances of the same element are allowed.
Let $A_i$ be a set of integers and let $\{A_i\}$ be the set of all sets $A_i$. Suppose set $A_i$ is given by $A_i = \{a | M_{ai} > 0\}$, where $M$ is a matrix. In other words, set $A_i$ contains the row indices $a$ of all elements in column $i$ of matrix $M$ which are greater than 0.
- How does one define the "inverse" case, i.e. $\{I_a\}$, from $\{A_i\}$ in set-builder notation?
Example:
Suppose
$$ M = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \\ \end{pmatrix} $$
then
$A_1 = \{1, 2, 3\}$, $A_2 = \{\}$, $A_3 = \{1, 3\}$
and
$I_1 = \{1, 3\}$, $I_2 = \{1\}$, $I_3 = \{1, 3\}$.
My idea would be: $I_a = \{i | \exists A_i \in \{A_i\} : a \in A_i\}$. Suppose $a = 1$ then $I_1 = \{i | \exists A_i \in \{A_i\} : 1 \in A_i\}$.
- What is the total number of elements in all sets in a set $\{A_i\}$ called (in this case 5) and is there a notation convention similar to the cardinality of a set? The cardinality of $\{A_i\}$ would be 3 but that's not what I am looking for.