And what does ⊂ mean when applied to two structures? After all, structures are not just sets; they are tuples consisting of the domain together with functions and/or relations. Therefore, the regular concept of set inclusion cannot be applied to structures, and a specific definition is needed.If N is a structure, then M ⊂ N means that M is a substructure of N.
According to Wikipedia, M is a substructure of N if the domain of M is a subset of the domain of N and the relations of M are the corresponding relations of N restricted to the domain of M. A subgraph is a not necessarily a substructure because it may fewer edges than what is inherited from the large graph. Subgraphs are called weak substructures. So, a relevant question about this problem is, are we talking about regular or weak substructures? Or what is the definition of a substructure?
I have no idea how that relates to mathematical logic. You seem to refuse to answer a simple question about the text material. If you tell us about the textbook, one of us may have a better idea as to what you are doing. Until then, I done with it.