This problem is about first order logic.
Given a structure A and a bijection g with domain |A| (i.e. the universe of A), I have to show that there is a unique structure B such that g is an isomorphism of A onto B.
Since A is isomorphic to itself, this basically comes down to showing that every structure is unique up to isomorphism. While I would feel somewhat comfortable showing this for certain types of sets, I'm having trouble seeing how to go about doing this with structure.
Any help would be appreciated.