If we have K_n denote the complete graph on n vertices, can any one explain to me how to know how many substructures does K_n have?
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?
If we have $\displaystyle N$ vertices then $\displaystyle \mathcal{K}_N$ has $\displaystyle \dbinom{N}{2}$ edges.
There are $\displaystyle \displaystyle 2^{\binom{N}{2}}$ possible subgraphs of $\displaystyle \mathcal{K}_N$.
(That counts the empty-edge subgraph).
Would tell us what textbook you are following?
I answered a Graph Theory problem.
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.