D-Separation in a Bayesian Network

Question

I am working with the graph below:

I know the three cases of d-separation are below (taken from here):

I need to find ALL pairs of nodes separated by {A} and {A, F}

My thought is:

{A} = [ {B, C} because of rule 1, {B, D} because of rule 1, [D, C} because of rule 1 ]

{A, F} = [ {B, C} because of rule 1, {B, D} because of rule 1, [D, C} because of rule 1 ]

I don't think {A, F} is correct, but I can't see why not, since {F} is just a leaf node and doesn't d-separate anything?

Answer 1

{A} = [ {B, C} because of rules 1 and 3, {D, C} because of rules 1 and 3 ]

For a pair of nodes to be d-separated by a set of nodes $S$, all paths between these nodes should be blocked by S. That's why {B, D} are not d-separated by { A } - there is a direct path between them that is not blocked by $A$.

{B, C} are indeed d-separated by {A}, but to see that we need to also consider the path $B \rightarrow E \leftarrow C$ which is blocked by case 3. The same goes for {D, C} and the path $D \leftarrow B \rightarrow E \leftarrow C$, which is again blocked by case 3 beacuase of $E$.

Now, these last two are actually not blocked by {A, F} Because $F$ is a descendant of $E$ and thus unblocks the above paths. Case 3 in your image actually extends to the descendants of E, as well as the node itself, not being observed.

So : {A, F} = []