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{\normalsize CS513:} \hfill {\bf hw4Template} \hfill {Fall, 2017}}

\jbf{1.} Let $\Sigma$ be an arbitrary vocabulary, $\A,\B\in\struc{\Sigma}$ be arbitrary and assume
that $\eta$ is an isomorphism from $\A$ to $\B$.

\answer{\jbf{1}. We show by induction on $\phi\in\sel(\Sigma)$ that $\A\models \phi
  \Leftrightarrow \B \models \phi$.}

\newpage
\jbf{2.} Let  $S\subseteq \N$ be arbitrary.
\answer{\jbf{2a.} Assume that $S = \sel(M)$ for some T.M., $M$ and assume that $S\ne \emptyset$.
  Construct a total TM computing the function $f$ s.t. $S = \set{f(0),f(1), f(2), \ldots}$.}


\vspace*{.1in}

\answer{\jbf{2b.} Assume that $S = \emptyset$ or  $S = \set{f(0),f(1), f(2), \ldots}$, for some
  total, recursive function $f$.  Construct a TM, $M$, s.t., $S = \sel(M)$.}

\newpage
\answer{\jbf{3a.} Equivalent or not?  $\tE\tF \phi \lor \tE\tF\psi; \quad \tE\tF (\phi \lor \psi)$}

\vspace*{.1in}

\answer{\jbf{3b.} Equivalent or not?    $\tA\tF \phi \lor \tA\tF\psi;\quad \tA\tF(\phi\lor\psi)$   }

\vspace*{.1in}



\answer{\jbf{3c.} Equivalent or not?     $\tA[p \tU \tA[q\tU r]];\quad \tA[\tA[p\tU
    q]\tU r]$  }

\newpage
\jbf{4.} Show that $\alpha \tR \beta \sE \lnot(\lnot \alpha \tU \lnot \beta)$ \; and \;
$\alpha \tU \beta \sE \lnot(\lnot \alpha \tR \lnot \beta)$. \quad  Let $\K$ be an arbitrary Kripke
structure and $\pi$ an arbitrary infinite path through $\K$.  Let $p,q \in \pvar$.


\answer{\jbf{4a.} Assume that $(\K,\pi) \models p\tR q$ and show that  $(\K,\pi) \models
  \lnot(\lnot p \tU \lnot q)$.}

\vspace*{.1in}

\answer{\jbf{4b.} Assume that $(\K,\pi) \models \lnot(\lnot p\tU \lnot q)$ and show that  $(\K,\pi) \models
   p \tR  q$.}


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