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\newcommand\cin{c_{\operatorname{in}}}
\newcommand\cout{c_{\operatorname{out}}}

\begin{document}

\setcounter{chapter}{2}
\chapter{Logic}

\begin{solution}{3.13}
  \begin{solnlist}
  \item The truth table for the full adder is as follows:
    \begin{center}
      \begin{tabular}{ccc|cc}
        $x$&$y$&$\cin$&$\cout$&$s$\\
        \hline
        0&0&0&0&0\\
        0&1&0&0&1\\
        1&0&0&0&1\\
        1&1&0&1&0\\
        0&0&1&0&1\\
        0&1&1&1&0\\
        1&0&1&1&0\\
        1&1&1&1&1\\
      \end{tabular}
    \end{center}
  \item Here is the truth table for the conjectured expression
    for~$\cout$: 
    \begin{center}
      \begin{tabular}{ccc|cc|ccc}
        $x$&$y$&$\cin$&$x\land y$&
        $\cin\land(x\oplus y)$&
        $(x\land y)\lor(\cin\land(x\oplus y))$\\
        \hline
        0&0&0&0&0&0\\
        0&0&1&0&0&0\\
        0&1&0&0&0&0\\
        0&1&1&0&1&1\\
        1&0&0&0&0&0\\
        1&0&1&0&1&1\\
        1&1&0&1&0&1\\
        1&1&1&1&0&1
      \end{tabular}
    \end{center}
    Here is the truth table for the conjectured expression for~$s$:
    \begin{center}
      \begin{tabular}{ccc|cc}
        $x$&$y$&$\cin$&$x\oplus y$&
        $x\oplus y\oplus \cin$\\ 
        \hline
        0&0&0&0&0\\
        0&0&1&0&1\\
        0&1&0&1&1\\
        0&1&1&1&0\\
        1&0&0&1&1\\
        1&0&1&1&0\\
        1&1&0&0&0\\
        1&1&1&0&1
      \end{tabular}
    \end{center}
  \end{solnlist}
\end{solution}  

\end{document}

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