In this picture you can see the hesse diagrem of $\subseteq$ over $P(\{x,y,z\})$
For the set $A$ with $k$ elements, $k>0$
look at the diagram as a graph, it's vertices are the members of $P(A)$
Show the graph is a regular graph (all vertices have the same degree)
what is the degree?
I know the answer is $k$, I just don't know how to prove it, please help.
Each set $S\subseteq A$ is adjacent "upwards" to the sets having one extra element which isn't in $S$. And it is adjacent "downwards" to the sets having one element of $S$ removed. So $S$ is adjacent one way or the other to one set for each element of $A$, whether or not it is an element of $S$. That is, every set is adjacent to $k$ other sets.