\problemdata                                                             
        {Pizza, Monkeys, and TAs} % latex-friendly label for the prob.
        {Quantifiers} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S01 Quiz1, 1} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Consider the following predicates where the universe of discourse is
the set of all living things:
\begin{description}
\item[$P(x)$:] $x$ is a monkey
\item[$Q(x)$:] $x$ is a 6.042 TA
\item[$R(x)$:] $x$ comes from the 23rd century
\item[$S(x)$:] $x$ likes to eat pizza
\end{description}
Express each of the following statements using quantifiers, logical
connectives, and and the above predicates:
\begin{problemparts}
\problempart No monkeys like to eat pizza.

\solution{\medskip
\textbf{Solution:}
$(\forall x) (P(x) \implies \neg S(x))$}

 
\problempart    Nobody from the 23rd century dislikes eating pizza.

\solution{\medskip
\textbf{Solution:}
$(\forall x) (R(x) \implies S(x))$}

 
\problempart    All 6.042 TAs are monkeys.

\solution{\medskip
\textbf{Solution:}
$(\forall x) (Q(x) \implies P(x))$}

 
\problempart    No 6.042 TA comes from the 23rd century.

\solution{\medskip
\textbf{Solution:}
$(\forall x) (Q(x) \implies \neg R(x))$}

 
\problempart Does (d) follow logically from (a), (b), and (c)?  If so, give
a proof.  If not, give a counterexample.

\solution{\medskip
\textbf{Solution:}

\begin{center}
\begin{tabular}{ll}
        $Q(x) \implies P(x)$ & from (c) \\
        $P(x) \implies \neg S(x)$ & from (a) \\
        $\neg S(x) \implies \neg R(x)$ & from contrapositive of (b)\\ 
        $\therefore Q(x) \implies \neg R(x)$
\end{tabular}

\end{center}

}

  
\problempart    Translate into English:
$(\forall x) (R(x) \vee S(x) \implies Q(x))$.

\solution{\medskip
\textbf{Solution:}
Anyone who is from the 23rd century or  likes to eat pizza is a 6.042 TA.}

 
\problempart     Translate into English: $(\exists x) (R(x) \wedge
\neg Q(x)) \implies (\forall x)(P(x) \implies S(x))$.

\solution{\medskip
\textbf{Solution:}
If there is a person from the 23rd century who is not a 6.042 TA then all monkeys like to eat pizza}

\solution{
Some students forgot leading quantifiers in the first parts (especially
$\forall x$). On the last part, many students mistakenly said that
the entity mentioned in the first `if' would be a monkey that liked
pizza. Note that this is not the same as the correct statement.
}

 
\end{problemparts}
\end{problem}
                                 