\documentclass[a4paper,10pt,draft]{article}
\usepackage{amsmath}
\input{../../math/abbreviations}
\input{../../math/format}

\title{First-order logic exercises}
\author{Math 406}
\date{2004.11.02}

\newtheorem{problem}{Problem}
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}

\begin{document}
\maketitle
\thispagestyle{empty}
  \begin{problem}
    Letting $P$ and $Q$ be unary predicates, determine, from the
    definition of $\models$, whether the following hold:
    \begin{enumerate}
      \item
$\Exists xPx\to \Exists xQx\models\Forall x(Px\to Qx)$;
\item
$\Forall xPx\to\Exists xQx\models\Exists x(Px\to Qx)$;
\item
$\Exists x(Px\to Qx)\models\Forall xPx\to\Exists xQx$.
    \end{enumerate}
  \end{problem}

  \begin{problem}
Let $\lang=\{R\}$, where $R$ is a binary predicate, and let $\str A$
be the $\lang$-structure $(\Z,\leq)$.  Determine $\phi^{\str A}$ if
$\phi$ is:
\begin{enumerate}
  \item
$\Forall {x_1}(Rx_1x_0\to Rx_0x_1)$;
\item
$\Forall {x_2}(Rx_2x_0\lor Rx_1x_2)$.
\end{enumerate}
  \end{problem}

  \begin{problem}
    Let $\lang$ be $\{S,P\}$, where $S$ and $P$ are binary
    function-symbols.  Then $(\R,+,\cdot)$ is an $\lang$-structure.
    Show that the following sets and relations are definable in this
    structure:
    \begin{enumerate}
            \item
$\{0\}$;
\item
$\{1\}$;
\item
$\{a\in\R:0< a\}$;
\item
$\{(a,b)\in\R^2:a<b\}$.
    \end{enumerate}
  \end{problem}

  \begin{problem}
    Show that the following sets are definable in
    $(\varN,+,\cdot,\leq,0,1)$:
    \begin{enumerate}
      \item
the set of even numbers;
\item
the set of prime numbers.
    \end{enumerate}
  \end{problem}

\end{document}