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%
% Proof Systems, etc.
%
\def\PL{{\mathrm{PL}}}
\def\PA{{\mathrm{PA}}}

%
% Logical syntax and symbols
%
\def\liff{\leftrightarrow}
\def\limplies{\rightarrow}
\def\tautEq{\vDash\Dashv}
\def\True{{\mathrm T}}
\def\False{{\mathrm F}}

\def\bigand{\bigwedge}
\def\bigor{\bigvee}

% Other symbols
\def\pprime{{\prime\prime}}
\def\mathfrakA{{\mathfrak A}}
\def\mathfrakB{{\mathfrak B}}
\def\mathfrakC{{\mathfrak C}}

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%
% Text in mathematics
%
\def\FmlaName{\mathit}
\def\ConnectiveName{\mathit}
\def\TBook{{\FmlaName{Book}}}
\def\TLikes{{\FmlaName{Likes}}}
\def\TJazz{{\FmlaName{Jazz}}}
\def\TKpop{{\FmlaName{Kpop}}}
\def\TKnows{{\FmlaName{Knows}}}
\def\TJoan{{\FmlaName{Joan}}}
\def\TMother{{\FmlaName{Mother}}}
\def\TPrime{{\FmlaName{Prime}}}
\def\TPrimePower{{\FmlaName{PrimePower}}}
\def\TRead{{\FmlaName{Read}}}
\def\TDog{{\FmlaName{Dog}}}
\def\TCat{{\FmlaName{Cat}}}
\def\TPerson{{\FmlaName{Person}}}
\def\TTime{\FmlaName{Time}}
\def\Ptime{{\mathrm P}}
\def\NP{{\mathrm{NP}}}
\def\TCase{{\ConnectiveName{Case}}}
\def\TMaj{{\ConnectiveName{Maj}}}


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\begin{document}


\begin{center}
{\bf Math 160A - Fall 2021 -
\bf Homework 6 - Due Wednesday, November 10, 10:00pm} \\
(Hand in by uploading to Gradescope)
\end{center}

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\noindent
{\bf 1.} Use unary predicates $\TJazz(x)$ and $\TKpop(x)$, the binary predicates
$\TLikes(x,y)$ and $\TKnows(x,y)$, the constant symbols \hbox{\it Joan} and \hbox{\it John},
and the equality sign~$=$
to express the following English sentences in first-order logic.  $\TJazz(x)$ means ``$x$~is a jazz musician'',
$\TKpop(x)$ means ``$x$~is a K-pop musician'', $\TLikes(x,y)$ means ``$x$ likes $y$'', and
$\TKnows(x,y)$ means ``$x$ knows $y$''. Variables
range over the universe of people.

Express the following as first-order formulas:

\kern -1.0em

\begin{enumerate}[nosep]
\item Jazz musicians do not like K-pop musicians.
\item Joan knows a jazz musician who likes every K-pop musician.
\item John likes everyone he knows.
\item There is no one that John and Joan both like.
\item Everyone that John likes knows a jazz musican.
\end{enumerate}

\noindent
{\bf 2.} Use the same language as above. For each sentence
(a)-(d), either: (i)~if it is a syntactically correct formula, give a translation into English,
 or (ii) state that it is not a syntactic
first-order formula.

\kern -1.0em

\begin{enumerate}[nosep]
\item $\forall x\, \exists y \, Likes(x, Jazz(y))$.
\item $\forall x\, (Likes(John,x) \liff Likes(Joan,x))$.
\item $\forall x\, \forall x\,Likes(Jazz(x), Jazz(y))$.
\item $\forall x\, (Likes(John,x) \limplies \exists y\,(Jazz(y) \land Likes(x, y)))$.
\end{enumerate}

\noindent
{\bf 3.} Show that none of (a), (b) or (c) is logically implied by
the other two. Do this by giving structures~$\mathfrak A$ which satisfy
only two of the three sentences.

\kern -1.0em

\begin{enumerate}[nosep]
\item $\forall x\, \forall y\, \forall z \, ( x \neq y \limplies (P(x,y) \liff \lnot P(y,x))$.
\item $\exists x \, \forall y \, (\lnot P(x,y))$.
\item $\forall x\, \forall y\, \forall z\, (P(x,y) \limplies P(y,z) \limplies P(x,z))$.
\end{enumerate}

\noindent
{\bf 4.} Show that $\forall y\, E(x,y)  \land \lnot E(z,x)$
is satisfiable by explicitly describing a pair $(\mathfrak A, \sigma)$ that satisfies it.

\noindent
{\bf 5.} Which of the following are tautologies?

\kern -1.0em

\begin{enumerate}[nosep]
\item $P(x,x) \limplies \exists y\, (P(y,y) \lor P(x,x))$.
\item $\forall x\, \forall y\, (P(x,y) \limplies P(x,y) \limplies P(x,y))$.
\item $P(x,y) \limplies P(x,y) \limplies P(x,y)$.
\end{enumerate}

\noindent
{\bf 6.}

\kern -1.0em

\begin{enumerate}[nosep]
\item Give an example of a formula $A$ so that $A \nvDash \forall x A$.
Give an example of pair $(\mathfrak A, \sigma)$ which demonstrates that
$A \nvDash \forall x A$.
\item Suppose that $x$ does not appear free in the formula~$A$.
Prove that $A \vDash \forall x A$.
\end{enumerate}

\end{document}
