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\noindent
\parbox{2.25in}{\bf Math 2110}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Sept. $11^\mathrm{th}$, 2020}

\vspace{0.2in}

{\bf\Large NOTE:} You are always welcome to type up your homework. However, you are not required to type up any of this particular assignment. You can turn in (neatly) handwritten work.

\vspace{0.2in}

\begin{enumerate}
\item Consider the statement $A(f(\underline{c})) \to ((\exists x\, \forall y\, B(x,y)) \wedge (\exists z\, C(z)))$.

We use a model where the universe is all people, $A(x)$ is defined by ``$x$ does not age'', $B(x,y)$ is defined by ``$x$ is always excellent towards $y$'', $C(x)$ is defined by ``$x$ parties on forever'', $\underline{c}$ is ``Ava Syme-Reeves'', and $f(y)$ is defined to be ``the father of $y$''. 

Using the above model, translate our statement to English. 

{\it Note:} Don't try to evaluate whether the statement is true or not.  It may or may not make sense.

\item Consider the statement $\forall x\,(A(x,f(x,\underline{c})) \wedge B(x,y)) \to \exists z\,(C(w,z) \vee D(z))$.

\begin{enumerate}[(a)]
\item Underline the scope of each quantifier.
\item Indicate which variables are bound and which ones are free. Is this a sentence? (Why or why not?)
\item Consider the term $t=g(x,\underline{a})$. Is $t$ free for $y$ in the above statement? (Why or why not?)
\item Consider the term $t=g(y,\underline{a})$. Is $t$ free for $y$ in the above statement? (Why or why not?)
\end{enumerate}

\item Consider the sentence $\forall x\, \exists y\, (A(x,y) \to (B(f(y)) \wedge C(\underline{a})))$.

\begin{enumerate}[(a)]
\item  Build a model (using $\mathbb{N} = \{0,1,2,\dots\}$ as the universe) where this sentence is true.
\item  Build a model (using $\mathbb{N} = \{0,1,2,\dots\}$ as the universe) where this sentence is false.
\item Is this sentence logically valid? (Why or why not?)
\end{enumerate}

\end{enumerate}

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