I'm trying to solve the following problem, Let A and B be sets with injections f:A→B and g:B→A. Fix a∈A and define [a] ={x∈A | ∃n∈ℕ s.t. (g◦f)n(a) =x or (g◦f)n(x) =a}.
I'd want to show that the various sets of the form [a] for a ∈ A give a partition of A. I'd also want to offer an example of how [a] may be a finite set even though A and B are infinite.
I thought that if I show that A ≠ B, so A∩B = ∅ and showing that there wouldn't be any empty subsets because the function is not necessarily surjective, so we have to exclude empty set, would help me to demonstrate that the different sets of [a] for a ∈ A result in an A partition; however, I don't think I can show this with only this information. Could someone help me on how to proceed for solving this question? All help is appreciated.
Note: Btw, by having a exponent in the composite function indicates that for example for the (g◦f)n(a) =x, it means that (g◦f)2 (a) =(g◦f◦g◦f)(a) = x and (g◦f)3(a) =(g◦f◦g◦f◦g◦f)(a) = x