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\usepackage{hw}

\begin{document}

\dotitle{4}{Thursday, October 5}

\begin{enumerate}


\item 
\begin{enumerate}

%\item Do problems 5 and 6 on page 28. In other words, show that
%  ``nand'' and ``nor'' are the only two complete binary connectives.
  
\item Show that $\{ \limplies, \bot \}$ is a complete set of
  connectives.

\item Show that $\{ \limplies, \lor, \land \}$ is not a  complete set of connectives.
  
\item Conclude that $\{ \limplies, \lor, \land, \liff, \top \}$ is not
  a complete set of connectives.  (Hint: define the last
  two in terms of the others.)

%\item Show that $\{ \bot, \liff \}$ is not a complete set of
%  connectives. (Hint: show that any formula involving only these
%  connectives and the variables $p_0$ and $p_1$ is equivalent to one
%  of the following: $\bot$, $\top$, $p_0$, $p_1$, $\lnot p_0$, $\lnot
%  p_1$, $p_0 \liff p_1$, or $p_0 \oplus p_1$.)
%\item Conclude that $\{ \bot, \top, \lnot, \liff, \oplus \}$ is not
%  complete. (Hint: see the previous problem.)
\end{enumerate}

\item Do problems 8, 10, and 11 on pages 28--29 of van Dalen.


\item \addstar How many ternary (3-ary) complete connectives are there?
  
%\item Do problem 10 on pages 28--29 of van Dalen.
  

\end{enumerate}


\end{document}