\item \self Given a state space of size $|S| = 2048$, find a symbolic binary encoding for this state space and compute the characteristic function for the sets of states $$B = \{s_0, s_1, s_2, ..., s_{1023}\} \enspace \text{and} \enspace C = \{s_{512}, s_{513},, s_{514}, ..., s_{1535}.\}$$ Then compute the characteristic function for the sets $D = B \cup C$ and $E = B \setminus C$. If possible, simplify the formulas.