Assume there exists at least a set (namely the empty set: $\emptyset$) and assume also that you have defined just one binary operation $P$ on any collection of sets (including those formed through the operation itself): $$ P(A, B) = \begin{cases} \{A, B\} & \text{if } A \neq B \\ \{A\} & \text{if } A=B \end{cases} $$ where $\{A, B\}$ denotes an unordered pair and $\{A\}$ denotes a singleton.
Now, my questions are:
- How many sets can I construct using only the operation $P$?
It is clear that the answer is infinitely many, however then it remains to clarify if it's a countable or uncountable infinity we are talking about;
- Is there the possibility that applying P in different ways could yield the same sets?
I do not even know how to start answering this one.
As always, any hint or comment is highly appreciated and let me know if I can explain my self clearer!