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+\subsection*{1.a}
+\emph{Consider the train gate controller example of slide 9, lecture
+\emph{Timed Automata}. Give an example of a series of timed transitions
+(including intermediate states) of the composed system (so the product
+construction of the three automata) showing a train approaching and finally
+leaving the gate.}
+\begin{figure}[ht]
+       \centering
+       \includegraphics[width=.3\linewidth]{1a}
+       \caption{Timed transitions of the composed system}
+\end{figure}
+
+\subsection*{1.b}
+\emph{Consider the timed automaton in figure 1 of the paper ”Timed Automata” by
+Rajeev Alur. Suppose initially we have a zone $(s0, [0\leq x\leq 4, 0\leq y
+\leq 3])$. Give the zone after a sequence a.b and show the intermediate steps
+in the derivation.}
+
+We define:\\
+$e_0=\langle s_0, a, \emptyset, [x:=0] s_1\rangle$ and\\
+$e_1=\langle s_1, a, \emptyset, [y:=0] s_2\rangle$\\
+
+And we derive:\\
+\scalebox{.99}{\parbox{.5\linewidth}{%
+       \begin{align*}
+               succ(e_1, succ(e_0, [0\leq x\leq 4, 0\leq y])) =&
+                       succ(e_1, ((([0\leq x\leq 4, 0\leq y]\wedge\emptyset)\Uparrow)
+                               \wedge\emptyset\wedge [x<1])[x:=0])\\
+               =& succ(e_1, (([0\leq x\leq 4, 0\leq y]\Uparrow)
+                       \wedge\emptyset\wedge [x<1])[x:=0])\\
+               =& succ(e_1, [0\leq x<1, 0\leq y][x:=0])\\
+               =& succ(e_1, [x=0, 0\leq y])\\
+               =& ((([x=0, 0\leq y]\wedge [x<1])\Uparrow)
+                       \wedge [x<1]\wedge [x<1])[y:=0]\\
+               =& (([0\leq x<1, 0\leq y]\Uparrow)\wedge [x<1]\wedge [x<1])[y:=0]\\
+               =& [0\leq x<1, y=0]\\
+       \end{align*}}}
+
+\subsection*{1.c}
+\emph{Consider the timed automaton in figure 1 of the paper \emph{Timed
+Automata} by Rajeev Alur. Give the zone automaton of the timed automaton, with
+initial state $(s0, [x = 0, y = 0])$.}
+\begin{figure}[ht]
+       \centering
+       \includegraphics[width=.7\linewidth]{1c}
+       \caption{Zone automaton}
+\end{figure}
+