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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}]
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{Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}}
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\label{ccis0}
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\vworkexercisechapterheader{}
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\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
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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
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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
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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
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$u=-2^{m-1}$, $u=\sim{}u+1$, and clearly a non-zero 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
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actual values of the integers represented.
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We present a second method now. Assume that $u \neq -2^{m-1}$, since
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$-2^{m-1}$ 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
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$u_{[m-1]}=1$, and by
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(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}),
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\end{vworkexercisesolution}
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\vworkexercisechapterfooter
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vfill
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\noindent\begin{figure}[!b]
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\noindent\rule[-0.25in]{\textwidth}{1pt}
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\begin{tiny}
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\begin{verbatim}
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$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 $
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$Revision: 1.3 $
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$Author: dtashley $
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$Date: 2002/09/12 23:30:20 $
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\end{verbatim}
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\end{tiny}
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\noindent\rule[0.25in]{\textwidth}{1pt}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% $Log: c_cis0.tex,v $
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% Revision 1.3 2002/09/12 23:30:20 dtashley
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% Safety checkin as changing machines.
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%
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% Revision 1.2 2002/08/26 17:57:03 dtashley
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% Additional solutions chapter added. Precautionary checkin to be sure
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% that I've captured all changes.
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%
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% Revision 1.1 2002/08/26 17:35:06 dtashley
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% Initial checkin.
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%
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%End of file C_CIS0.TEX
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