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Revision as of 03:12, 8 June 2008
, 03:12, 8 June 2008→Appendices @ PlanetMath : TeX Format: update
=Appendices @ PlanetMath : TeX Format=
=Appendices @ PlanetMath : TeX Format=
<pre>
\subsection{Table A1. Propositional Forms on Two Variables}
Table A1 lists equivalent expressions for the boolean functions of two variables in a number of different notational systems.
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{Table 1. Propositional Forms on Two Variables} \\
\multicolumn{7}{c}{Table A1. Propositional Forms on Two Variables} \\
\hline
\hline
$\mathcal{L}_1$ &
$\mathcal{L}_1$ & $\mathcal{L}_2$ &&
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ & $\mathcal{L}_4$ &
$\mathcal{L}_3$ &
$\mathcal{L}_5$ & $\mathcal{L}_6$ \\
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $x =$ & 1 1 0 0 & & & \\
\hline
\hline
\end{tabular}\end{quote}
\end{tabular}\end{quote}
\subsection{Table A2. Propositional Forms on Two Variables}
Table A2 lists the sixteen boolean functions of two variables in a different order, grouping them by structural similarity into seven natural classes.
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|}
\multicolumn{7}{c}{Table 2. Propositional Forms on Two Variables} \\
\multicolumn{7}{c}{Table A2. Propositional Forms on Two Variables} \\
\hline
\hline
$\mathcal{L}_1$ &
$\mathcal{L}_1$ & $\mathcal{L}_2$ &&
$\mathcal{L}_2$ &&
$\mathcal{L}_3$ & $\mathcal{L}_4$ &
$\mathcal{L}_3$ &
$\mathcal{L}_5$ & $\mathcal{L}_6$ \\
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $x =$ & 1 1 0 0 & & & \\
\hline
\hline
$f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\
$f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\
\hline
\end{tabular}\end{quote}
\subsection{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}
\begin{quote}\begin{tabular}{|c|c||c|c|c|c|}
\multicolumn{6}{c}{Table A3. $\operatorname{E}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$} \\
\hline
& &
$\operatorname{T}_{11}$ &
$\operatorname{T}_{10}$ &
$\operatorname{T}_{01}$ &
$\operatorname{T}_{00}$ \\
& $f$ &
$\operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y}$ &
$\operatorname{E}f|_{\operatorname{d}x (\operatorname{d}y)}$ &
$\operatorname{E}f|_{(\operatorname{d}x) \operatorname{d}y}$ &
$\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\
\hline
$f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\
\hline
$f_{1}$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ \\
$f_{2}$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ \\
$f_{4}$ & $x\ (y)$ & $(x)\ y$ & $(x)(y)$ & $x\ y$ & $x\ (y)$ \\
$f_{8}$ & $x\ y$ & $(x)(y)$ & $(x)\ y$ & $x\ (y)$ & $x\ y$ \\
\hline
$f_{3}$ & $(x)$ & $x$ & $x$ & $(x)$ & $(x)$ \\
$f_{12}$ & $x$ & $(x)$ & $(x)$ & $x$ & $x$ \\
\hline
$f_{6}$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ \\
$f_{9}$ & $((x,\ y))$ & $((x,\ y))$ & $(x,\ y)$ & $(x,\ y)$ & $((x,\ y))$ \\
\hline
$f_{5}$ & $(y)$ & $y$ & $(y)$ & $y$ & $(y)$ \\
$f_{10}$ & $y$ & $(y)$ & $y$ & $(y)$ & $y$ \\
\hline
$f_{7}$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ \\
$f_{11}$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ \\
$f_{13}$ & $((x)\ y)$ & $(x\ (y))$ & $(x\ y)$ & $((x)(y))$ & $((x)\ y)$ \\
$f_{14}$ & $((x)(y))$ & $(x\ y)$ & $(x\ (y))$ & $((x)\ y)$ & $((x)(y))$ \\
\hline
$f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\
\hline
\multicolumn{2}{c}{Fixed Point Total:} & 4 & 4 & 4 & 16 \\
\hline
\hline
\end{tabular}\end{quote}
\end{tabular}\end{quote}
