Bug with follow logs

The logs for the follow sets add a wrong {$} set if the symbol has nothing after it and that's wrong

The grammar:

S -> A B
A -> a A | epsilon
B -> b B | b

Expected:

Follow(S) = Follow(S')            = {$}
Follow(A) = First(B) ∪ Follow(A)  = {b}
Follow(B) = Follow(S) ∪ Follow(B) = {$}

Actual:

Follow(S) = { $ } u Follow(S') = { $ }
Follow(A) = First(B) u { $ } u Follow(A) = { b }
Follow(B) = { $ } u Follow(S) u Follow(B) = { $ }