Count number of sets containing an element

Question

I am looking for some terminology that illustrates counting the number of indexed sets (eg Si) containing a particular element (eg p).

Answer 1

My pleasure.

If $I$ is the index set of $(S_i)_{i \in I}$, then $|\{i \in I: p \in S_i\}|$ is the number (i.e., cardinal) of indexed sets $S_i$ containing $p$.