If we let f: A -> B be a function and A, B be two non-empty sets.
There exists a equivalence relation definied as
a ~ b ⇔ f(a) = f(b) for all a,b ∈ A
I am asked to describe the equivalence classes
My textbook says the answer is
Let A* $ =f^{-1}(y)$ then the equivalence classes corresponding to ~ are {A* : y ∈ B}.
I am having a bit of trouble understanding what the answer means. I can't really see how the f(a) = f(b) condition is being taken into consideration when constructing the equivalence classes.
Is the answer simply saying that the preimage of every element in B has it's own equivalence class in A? Would that mean that every equivalence class only consists of one element?
Thank you