I'm trying to figure out how to resolve this exercise. $$ \Sigma = \{a,b\} $$ is a set while $$ \mathcal{P}(\Sigma^*) $$ is the partition of sigma star. I have a function f: $$ f: \mathcal{P}(\Sigma^*) = \mathcal{P}(\Sigma^*)$$ such that for all subset P of sigma star, if P is not semidecidable and not decidable, then f(P) is a decidable set and $$ P \subseteq f(P) $$
Show that any such function f is not an injection.