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%
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%
% Text in mathematics
%
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\def\TLikes{{\FmlaName{Likes}}}
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\begin{document}


\begin{center}
{\bf Math 160B - Winter 2022 -
\bf Homework 4} \\  {\bf Due Saturday, February 5, 11:00pm} \\
(Hand in by uploading to Gradescope)
\end{center}

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You can also find these problems in the PDF text.

\noindent
{\bf 1.}
Prove that every infinite c.e.\ set~$R$ has an infinite decidable subset~$S$.

\noindent
{\bf 2.}
Let $R$ be a set (a unary relation) and $f$~be a unary function.
The \emph{preimage}~$f^{-1}[R]$ of~$R$ under~$f$ is the set
$\{ x : f(x) \in R \}$.
\begin{enumerate}
\item Suppose $R$ is decidable and $f$~is computable.
Prove that $f^{-1}[R]$ is decidable.
\item Suppose $R$ is c.e.\ and $f$~is partial computable.
Prove that $f^{-1}[R]$ is c.e.
\end{enumerate}

\noindent
{\bf 3.}
Let $R$ be a set (a unary relation) and $f$~be a unary function.
The \emph{image}~$f[R]$ of~$R$ under~$f$ is the set
$\{ f(x) : x \in R \}$.
Prove that there a decidable set~$R$ and a computable function~$f$
such that $f[R]$ is undecidable.

\noindent{\bf 4.}

\kern -30pt

\begin{enumerate}
\item
Prove that the sets $\THalt_0$, $\THalt_1$ and $\THalt_\TSelf$
are c.e.  You will need to use the assumption that there is
a universal algorithm~$U$: be sure to point out where
you use the universal algorithm, and explain why this is
needed. 
\item
Conclude that these three sets are
not co-c.e.
\end{enumerate}

\end{document}
