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.
this is a better link that explains what a structure is in the context of mathematical logic.
If I understand correctly, g is an isomorphism between A and B if
(1) g is a bijection between |A| and |B|,
(2) for every functional symbol f and every it is the case that , and
(3) for every predicate symbol R and every it is the case that iff .
The universe of B must be the image of g. We need to show that there exists one and only one interpretation of functional and predicate symbols on |B| such that g is an isomorphism. Well, this interpretation is uniquely determined by properties (2) and (3).
Thanks that actually helped, even though I was sort of missing the point and realized that the problem basically says: given bijection g there IS a UNIQUE structure B isomorphic to A UNDER g. Might sound redundant but it makes sense to me now.