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%
% Proof Systems, etc.
%
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%
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%
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\def\False{{\mathrm F}}

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

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\def\mathfrakC{{\mathfrak C}}

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\def\TLikes{{\FmlaName{Likes}}}
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\def\TPrime{{\FmlaName{Prime}}}
\def\TPrimePower{{\FmlaName{PrimePower}}}
\def\TRead{{\FmlaName{Read}}}
\def\TDog{{\FmlaName{Dog}}}
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\begin{document}


\begin{center}
{\bf Math 160A - Fall 2021 -
\bf Homework 5 - Due Wednesday, November 3, 10:00pm} \\
(Hand in by uploading to Gradescope)
\end{center}

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For problems 2 and 3, $L_\PA$ is the language $0, S, +, \cdot$.

\noindent
{\bf 1.} A quote that is sometimes (mis?)attributed to Abraham Lincoln is:
``You can fool some of the people all of the time. You can fool all
of the people some of the time. But you cannot fool all of the people all
of the time.''
Let these three assertions be:
\begin{enumerate}[itemsep=0pt]
\item You can fool some of the people all of the time.
\item You can fool all of the people some of the time.
\item You cannot fool all of the people all of the time.
\end{enumerate}
Express these three sentences in first-logic. Let variables
range over the universe that includes all people and all times.
Let $\TPerson(x)$ mean $x$ is a person, and $\TTime(x)$ mean $x$ is a time.
Let $F(x,t)$ mean you can fool person $x$ at time~$t$.

Sentences (a) and (b) have two possible translations to first-order
logic. Give both translations. Explain how the two translations
differ in meaning.
\bigskip

\noindent{\bf 2.} Express the following in $L_\PA$ (over the non-negative integers).
\begin{enumerate}[itemsep=0pt]
\item $x$ is less than $y$.
\item For any $n$ and $m$ with $m>0$, there are unique values
$p$ and~$r$ so that $n = q\cdot m + r$ and $r<m$. (Division algorithm.)
\end{enumerate}

\bigskip

\noindent{\bf 3.} Express the following in $L_\PA$ (over the non-negative integers).
\begin{itemize}[itemsep=0pt]
\item There are infinitely many primes.
\end{itemize}
There is no way to talk directly about ``infinitely many''. Instead, express this
by saying there are arbitrarily large primes.

\bigskip

\noindent{\bf 4.} Over the language with a unary predicate~$P(x)$ and
(as always) the equality sign~$=$, express the following in first-order logic.
\begin{enumerate}[itemsep=0pt]
\item There is a unique $x$ satisfying $P(x)$.
\item There are at least two objects~$x$ that satisfy $P(x)$.
\item There are at least three objects~$x$ that satisfy $P(x)$.
\item There are exactly two objects~$x$ that satisfy $P(x)$.
\end{enumerate}

\end{document}
