Let's say I have a multi-set: $(1,1,2)$ And I want to define a bijection from $A$ to $B$ such that:
$(1,1,2)$ becomes {3} and,
$(1,1,1)$ becomes {} and,
$(2,2,2)$ becomes {1,2,3} or,
in general, it is a set of the positions of $2$ in $A$.
So the rough idea of f is as below: f( (n_1, ... , n_some_number) ) = {"positions of 2 from n_i"}
But I am not so confident in defining this function in mathematical notion. So I would love to hear some advice.
Let me know if my question makes sense, I will add more info accordingly!
Multisets are not ordered, so this does not really make sense. You can, however, define a function from the set of finite sequences of positive integers to the set of finite sets of positive integers that sends $\langle n_1,\ldots,n_m\rangle$ to $\{k\in[m]:n_k=2\}$, where as usual $[m]=\{1,\ldots,m\}$. If you’re considering only $m$-tuples for some fixed $m$, then each will be sent to a subset of $[m]$, but the map is not injective and hence not a bijection.