\item \self Consider the propositional formula $\psi = (p \imp q)
    \land(q \imp r) \land (\neg r \lor p)$.

\begin{enumerate}
  \item Fill out the truth table for $\varphi$ (and its
      subformulas).

\begin{tabular}{|c|c|c||c|c|c|c|c|}
\hline
$p$&$q$&$r$&$(p \imp q)$&$(q \imp r)$&$\;\neg r\;$&$(\neg r \lor p)$&$\quad\psi\quad$\\
\hline
\hline
\textbf{F} &\textbf{F} &\textbf{F} & \T & \T & \T & \T & \T\\
\hline
\textbf{F} &\textbf{F} &\textbf{T} & \T & \T & \F & \F & \F\\
\hline
\textbf{F} &\textbf{T} &\textbf{F} & \T & \F & \T & \T & \F\\
\hline
\textbf{F} &\textbf{T} &\textbf{T} & \T & \T & \F & \F & \F\\
\hline
\textbf{T} &\textbf{F} &\textbf{F} & \F & \T & \T & \T & \F\\
\hline
\textbf{T} &\textbf{F} &\textbf{T} & \F & \T & \F & \T & \F\\
\hline
\textbf{T} &\textbf{T} &\textbf{F} & \T & \F & \T & \T & \F\\
\hline
\textbf{T} &\textbf{T} &\textbf{T} & \T & \T & \F & \T & \T\\
\hline
\end{tabular}

\item Is $\varphi$ satisfiable? \\
\quad Yes.
\item  Is $\varphi$ valid? \\
\quad No.
\item Give a formula $\varphi$ that semantically entails $\psi$. \\
\quad An equivalent formula $\varphi$ semantically entails $\psi$, therefore we let $\varphi = \psi$.
\item How can you check, using a truth table, whether $\varphi$
    semantically entails $\psi$? \\
\quad We can check this by looking at the models that satisfy the two formulas. If every model that satisfies $\varphi$ also models $\psi$, then $\varphi$ semantically entails $\psi$.

\end{enumerate}
