1 The switching circuit in Fig. 1.1 shows switches, \(s\) for a car's sidelights, \(h\) for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights. \begin{figure}[h] \end{figure} (Note: \(s\) and \(h\) are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)

1. 1. Describe in words the conditions under which the foglights will come on. Fig. 1.2 shows a combinatorial circuit. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{83d00f1f-25e5-4a26-bac5-dc2baf965438-02_367_1235_1183_431} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   2. Write the output in terms of a Boolean expression involving \(s , h\) and \(f\).
   3. Use a truth table to prove that \(\mathrm { s } \wedge \mathrm { h } \wedge \mathrm { f } = \sim ( \sim \mathrm { s } \vee \sim \mathrm { h } ) \wedge \mathrm { f }\).
2. A car's first gear can be engaged ( g ) if either both the road speed is low ( r ) and the clutch is depressed ( d ), or if both the road speed is low ( r ) and the engine speed is the correct multiple of the road speed (m).
   1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent \(r\), and single switches to represent each of \(\mathrm { d } , \mathrm { m }\) and g .
   2. Draw a combinatorial circuit to represent the Boolean expression \(\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g }\).
   3. Use Boolean algebra to prove that \(\mathrm { r } \wedge ( \mathrm { d } \vee \mathrm { m } ) \wedge \mathrm { g } = ( ( \mathrm { r } \wedge \mathrm { d } ) \vee ( \mathrm { r } \wedge \mathrm { m } ) ) \wedge \mathrm { g }\).
   4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch.