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\noindent
\parbox{2.25in}{\bf Math 2110-410}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Feb. $3^\mathrm{rd}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item Let  $\underline{c}$ be a constant, $f(x)$ be a function, and $P(x,y), Q(x)$ be predicates.
   \begin{enumerate}[(a)]
   \item Consider the sentence ``$\forall x \; (P(x,\underline{c}) \to Q(f(x)) $''.
         Show that this sentence is satisfiable (find a model for the sentence). Is this sentence logically valid? Prove your answer.
   \item Write a sentence (i.e. no free variables) involving $\underline{c}, f(x), P(x,y),$  and $ Q(x)$ which is unsatisfiable. Explain why your sentence is unsatisfiable.
   \item Write a sentence (i.e. no free variables) involving $\underline{c}, f(x), P(x,y),$ and $Q(x)$ which is satisfiable but not logically valid. Construct models showing that your sentence is satisfiable but not logically valid.
   \item Write a sentence (i.e. no free variables) involving $\underline{c}, f(x), P(x,y),$ and $Q(x)$ which is logically valid. Prove that your sentence is logically valid.
   \end{enumerate}

{\bf Note:} For parts (c), (d), and (e), don't reuse the sentence from part (a).
    
\item Prove Theorems K8, K10, K19, K25, K30, and K35   % did K1-K7, K9
\end{enumerate}

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