Given below are two finite state automata ( $\rightarrow$ indicates the start state and $F$ indicates a final state)
$$\overset{Y}{\begin{array}{|l|l|l|}\hline \text{} & \textbf{a} & \textbf{b} \\\hline \text{$\rightarrow$ $1$} & \text{1} & \text{2} \\\hline \text{$2 (F)$} & \text{2} & \text{1} \\\hline \end{array}} \qquad \overset{Z}{\begin{array}{|l|l|l|}\hline \text{} & \textbf{a} & \textbf{b} \\\hline \text{$\rightarrow$ $1$} & \text{2} & \text{2} \\\hline \text{$2 (F)$} & \text{1} & \text{1} \\\hline \end{array}}$$
Which of the following represents the product automaton $Z \times Y$?
- $\begin{array}{|l|l|}\hline \text{} & \text{a} & \text{b} \\\hline \text{$\rightarrow$ $P$} & \text{S} & \text{R} \\\hline \text{Q} & \text{R} & \text{S} \\\hline \text{R(F)} & \text{Q} & \text{P}\\\hline \text{S} & \text{Q} & \text{P}\\\hline \end{array}$
- $\begin{array}{|l|l|}\hline \text{} & \text{a} & \text{b} \\\hline \text{$\rightarrow$ $P$} & \text{S} & \text{Q} \\\hline \text{Q} & \text{R} & \text{S} \\\hline \text{R(F)} & \text{Q} & \text{P}\\\hline \text{S} & \text{P} & \text{Q}\\\hline \end{array}$
- $\begin{array}{|l|l|}\hline \text{} & \text{a} & \text{b} \\\hline \text{$\rightarrow$ $P$} & \text{Q} & \text{S} \\\hline \text{Q} & \text{R} & \text{S} \\\hline \text{R(F)} & \text{Q} & \text{P}\\\hline \text{S} & \text{Q} & \text{P}\\\hline \end{array}$
- $\begin{array}{|l|l|}\hline \text{} & \text{a} & \text{b} \\\hline \text{$\rightarrow$ $P$} & \text{S} & \text{Q} \\\hline \text{Q} & \text{S} & \text{R} \\\hline \text{R(F)} & \text{Q} & \text{P}\\\hline \text{S} & \text{Q} & \text{P}\\\hline \end{array}$