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%
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%
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\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
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\def\mathfrakC{{\mathfrak C}}

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%
% Text in mathematics
%
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\def\TDog{{\FmlaName{Dog}}}
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\begin{document}

\begin{center}
{\bf Math 160B - Winter 2022 -
\bf Quiz 4 - Upload by 5:00pm, Friday, January 21} \\
\end{center}

\kern -1em

\noindent
\begin{tabbing}
{\bf Start Time:} \hspace*{1.2in}\={\bf Your name:}\\[0.5ex]
{\bf Stop Time:} \> {\bf Integrity signature:}
\end{tabbing}

\vskip -1em

\noindent
Suggested time limit 15 minutes, not counting download and upload. Please add explanation if over 20 minutes.

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\noindent
{\bf Definition:} Recall that for $v$ and $w$ strings,
$v \circ w = vw$ is the concatenation of $v$ and~$w$.
Let the concatenation $R\circ S$ of two sets $R$ and $S$ of strings in $\Sigma^*$ be
the set of strings
\[
R\circ S ~=~ \{ v\circ w : \hbox{$v\in R$ and $w\in S$} \}.
\]
For example,$\{00,111\}\circ\{0, 1\} = \{ 000, 001, 1110, 1111\}$.

\noindent
{\bf 1.} Let $\Sigma^* = \{ 0,1\}$. Suppose $R$ and $S$ are subsets
of~$\Sigma^*$ (i.e., they are unary relations.)  Further suppose $R$
and~$S$ are computably enumerable.  Prove that $R\circ S$ is computably 
enumerable either by giving an algorithm that enumerates $R\circ S$
or by giving an algorithm that semidecides $R\circ S$.

\vfill


\end{document}
