\item \lect Consider the propositional formulas $\varphi = (p \rightarrow
    q) \vee \neg r$ and $\psi = (\neg r \wedge p) \vee (\neg q \rightarrow \neg r)$.

\begin{enumerate}
  \item Fill out the truth table for $\varphi$ and $\psi$ and
      their subformulas.

\begin{tabular}{|c|c|c||c|c|c|c|c||c|c|}
\hline
$p$&$q$&$r$&$\neg q$&$\neg r$&$p \rightarrow q$&$\neg r \wedge p$&$\neg q \rightarrow \neg r$&$\varphi$&$\psi$\\
\hline
\hline
\textbf{F} &\textbf{F} &\textbf{F} & T & T & T & F & T & T & T \\
\hline 
\textbf{F} &\textbf{F} &\textbf{T} & T & F & T & F & F & T & F \\
\hline
\textbf{F} &\textbf{T} &\textbf{F} & F & T & T & F & T & T & T \\
\hline
\textbf{F} &\textbf{T} &\textbf{T} & F & F & T & F & T & T & T \\
\hline
\textbf{T} &\textbf{F} &\textbf{F} & T & T & F & T & T & T & T \\
\hline
\textbf{T} &\textbf{F} &\textbf{T} & T & F & F & F & F & F & F \\
\hline
\textbf{T} &\textbf{T} &\textbf{F} & F & T & T & T & T & T & T \\
\hline
\textbf{T} &\textbf{T} &\textbf{T} & F & F & T & F & T & T & T \\
\hline
\end{tabular}
\vspace{0.5cm}

\item Which of the formulas is satisfiable?

Both of them are satisfiable.

\item Which of the formulas is valid?

None of them are valid.

\item Which of the two formulas $\varphi$ and $\psi$ entails the other?

It holds that $\psi \models \varphi$.

\end{enumerate}