Is a subgraph either a spanning subgraph or a full subgraph? A graph $G' = (N' ,A')$ is a spanning subgraph of a graph $G = (N, A)$
iff $N ' = N$ and $A' \subseteq A$.
A graph $G' = (N',A')$ is a full subgraph of a
graph $G = (N, A)$ iff $N' \subseteq N$ and $A' = \{(x,y) \in A \mid x,y \in N '\}$.


I have to prove that &quot;If $G'$ is a subgraph of a graph $G$, then either
G' is a spanning subgraph of $G$, or $G'$ is a full subgraph of $G$. If this statement
is valid, provide a proof that it is valid, otherwise provide a counterexample&quot; but isn't a full subgraph actually a spanning subgraph?

I mean that any subgraph of $G$ is a full subgraph in the end, right?
Any ideas how I can prove the statement?

My second question is how to calculate the number of spanning subgraphs of $G$ as a function of the number of nodes in $N$. How can I do this without knowing $A$ or how many edges I have in the graph?
On your question "isn't a full subgraph actually a spanning subgraph?". The answer is no, a full subgraph doesn't need to be a spanning subgraph. By your definition, a full subgraph can have lesser number of vertices than in the original graph. However, a spanning subgraph must have exactly the same set of vertices in the original graph.

On your second question "how to calculate the number of spanning subgraphs of $G$ as a function of number of nodes in $N$." Since it is a fact that graph $G = (N,A)$ with differing $A$ will have differing number of possible spanning subgraphs, I suspect you might have misunderstood the question. For example, does the question actually only ask for an upper bound of the number of possible spanning subgraphs as function of $|N|$?