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%$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v 1.3 2002/09/12 23:30:20 dtashley Exp $ 
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\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}] 
\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}] 
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{Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}} 
{Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}} 
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\label{ccis0} 
\label{ccis0} 
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\vworkexercisechapterheader{} 
\vworkexercisechapterheader{} 
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\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}} 
\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}} 
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We can show this result in two ways. The first way, based on bit patterns, is to note 
We can show this result in two ways. The first way, based on bit patterns, is to note 
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that adding an $m$bit number, $u$, to its one's complement will result in a bit pattern 
that adding an $m$bit number, $u$, to its one's complement will result in a bit pattern 
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containing all 1's, i.e. $\forall i$, $u_{[i]} = 1$. Adding 1 to this bit pattern will 
containing all 1's, i.e. $\forall i$, $u_{[i]} = 1$. Adding 1 to this bit pattern will 
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always produce $\forall i$, $u_{[i]} = 0$ with a carry out. Since the order of addition 
always produce $\forall i$, $u_{[i]} = 0$ with a carry out. Since the order of addition 
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does not matter, this establishes that adding $u$ to $\sim{}u+1$ will produce 0, thus showing 
does not matter, this establishes that adding $u$ to $\sim{}u+1$ will produce 0, thus showing 
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that $u$ and $\sim{}u+1$ are additive inverses. This method, although valid, does not 
that $u$ and $\sim{}u+1$ are additive inverses. This method, although valid, does not 
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establish that $u$ and $\sim{}u+1$ actually represent additive inverses. For example, if 
establish that $u$ and $\sim{}u+1$ actually represent additive inverses. For example, if 
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$u=2^{m1}$, $u=\sim{}u+1$, and clearly a nonzero number cannot be an additive inverse of 
$u=2^{m1}$, $u=\sim{}u+1$, and clearly a nonzero number cannot be an additive inverse of 
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itself. Thus, it would be more comforting to show this result in a way that demonstrates the 
itself. Thus, it would be more comforting to show this result in a way that demonstrates the 
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actual values of the integers represented. 
actual values of the integers represented. 
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We present a second method now. Assume that $u \neq 2^{m1}$, since 
We present a second method now. Assume that $u \neq 2^{m1}$, since 
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$2^{m1}$ cannot have an additive inverse in an $m$bit signed integer. 
$2^{m1}$ cannot have an additive inverse in an $m$bit signed integer. 
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If $u=0$, $\sim{}u+1=0$, so the relationship is clearly met. If $u<0$, then 
If $u=0$, $\sim{}u+1=0$, so the relationship is clearly met. If $u<0$, then 
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$u_{[m1]}=1$, and by 
$u_{[m1]}=1$, and by 
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(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}), 
(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}), 
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\end{vworkexercisesolution} 
\end{vworkexercisesolution} 
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\vworkexercisechapterfooter 
\vworkexercisechapterfooter 
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\vfill 
\vfill 
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\noindent\begin{figure}[!b] 
\noindent\begin{figure}[!b] 
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\noindent\rule[0.25in]{\textwidth}{1pt} 
\noindent\rule[0.25in]{\textwidth}{1pt} 
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\begin{tiny} 
\begin{tiny} 
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\begin{verbatim} 
\begin{verbatim} 
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$RCSfile: c_cis0.tex,v $ 
$RCSfile: c_cis0.tex,v $ 
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$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v $ 
$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v $ 
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$Revision: 1.3 $ 
$Revision: 1.3 $ 
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$Author: dtashley $ 
$Author: dtashley $ 
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$Date: 2002/09/12 23:30:20 $ 
$Date: 2002/09/12 23:30:20 $ 
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\end{verbatim} 
\end{verbatim} 
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\end{tiny} 
\end{tiny} 
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\noindent\rule[0.25in]{\textwidth}{1pt} 
\noindent\rule[0.25in]{\textwidth}{1pt} 
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\end{figure} 
\end{figure} 
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% $Log: c_cis0.tex,v $ 
% $Log: c_cis0.tex,v $ 
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% Revision 1.3 2002/09/12 23:30:20 dtashley 
% Revision 1.3 2002/09/12 23:30:20 dtashley 
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% Safety checkin as changing machines. 
% Safety checkin as changing machines. 
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% 
% 
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% Revision 1.2 2002/08/26 17:57:03 dtashley 
% Revision 1.2 2002/08/26 17:57:03 dtashley 
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% Additional solutions chapter added. Precautionary checkin to be sure 
% Additional solutions chapter added. Precautionary checkin to be sure 
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% that I've captured all changes. 
% that I've captured all changes. 
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% 
% 
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% Revision 1.1 2002/08/26 17:35:06 dtashley 
% Revision 1.1 2002/08/26 17:35:06 dtashley 
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% Initial checkin. 
% Initial checkin. 
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%End of file C_CIS0.TEX 
%End of file C_CIS0.TEX 