Talk:PlanetPhysics/Differential Propositional Calculus Appendix 1

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\textit{\textbf{Note.} The following Tables are best viewed in the Page Image mode.}

\tableofcontents

\subsection{Table A1. Propositional Forms on Two Variables}

Table A1 lists equivalent expressions for the \htmladdnormallink{boolean functions}{http://planetphysics.us/encyclopedia/Predicate.html} of two variables in a number of different notational systems.

\begin{quote}\begin{tabular}{|c|c|c|c|c|c|c|} \multicolumn{7}{c}{\textbf{Table A1. Propositional Forms on Two Variables}} \\ \hline $\mathcal{L}_1$ & $\mathcal{L}_2$ && $\mathcal{L}_3$ & $\mathcal{L}_4$ & $\mathcal{L}_5$ & $\mathcal{L}_6$ \\ \hline & & $x =$ & 1 1 0 0 & & & \\ & & $y =$ & 1 0 1 0 & & & \\ \hline $f_{0}$ & $f_{0000}$ && 0 0 0 0 & $(~)$ & $\operatorname{false}$ & $0$ \\ $f_{1}$ & $f_{0001}$ && 0 0 0 1 & $(x)(y)$ & $\operatorname{neither}\ x\ \operatorname{nor}\ y$ & $\lnot x \land \lnot y$ \\ $f_{2}$ & $f_{0010}$ && 0 0 1 0 & $(x)\ y$ & $y\ \operatorname{without}\ x$ & $\lnot x \land y$ \\ $f_{3}$ & $f_{0011}$ && 0 0 1 1 & $(x)$ & $\operatorname{not}\ x$ & $\lnot x$ \\ $f_{4}$ & $f_{0100}$ && 0 1 0 0 & $x\ (y)$ & $x\ \operatorname{without}\ y$ & $x \land \lnot y$ \\ $f_{5}$ & $f_{0101}$ && 0 1 0 1 & $(y)$ & $\operatorname{not}\ y$ & $\lnot y$ \\ $f_{6}$ & $f_{0110}$ && 0 1 1 0 & $(x,\ y)$ & $x\ \operatorname{not~equal~to}\ y$ & $x \ne y$ \\ $f_{7}$ & $f_{0111}$ && 0 1 1 1 & $(x\ y)$ & $\operatorname{not~both}\ x\ \operatorname{and}\ y$ & $\lnot x \lor \lnot y$ \\ \hline $f_{8}$ & $f_{1000}$ && 1 0 0 0 & $x\ y$ & $x\ \operatorname{and}\ y$ & $x \land y$ \\ $f_{9}$ & $f_{1001}$ && 1 0 0 1 & $((x,\ y))$ & $x\ \operatorname{equal~to}\ y$ & $x = y$ \\ $f_{10}$ & $f_{1010}$ && 1 0 1 0 & $y$ & $y$ & $y$ \\ $f_{11}$ & $f_{1011}$ && 1 0 1 1 & $(x\ (y))$ & $\operatorname{not}\ x\ \operatorname{without}\ y$ & $x \Rightarrow y$ \\ $f_{12}$ & $f_{1100}$ && 1 1 0 0 & $x$ & $x$ & $x$ \\ $f_{13}$ & $f_{1101}$ && 1 1 0 1 & $((x)\ y)$ & $\operatorname{not}\ y\ \operatorname{without}\ x$ & $x \Leftarrow y$ \\ $f_{14}$ & $f_{1110}$ && 1 1 1 0 & $((x)(y))$ & $x\ \operatorname{or}\ y$ & $x \lor y$ \\ $f_{15}$ & $f_{1111}$ && 1 1 1 1 & $((~))$ & $\operatorname{true}$ & $1$ \\ \hline \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|} \multicolumn{7}{c}{\textbf{Table A2. Propositional Forms on Two Variables}} \\ \hline $\mathcal{L}_1$ & $\mathcal{L}_2$ && $\mathcal{L}_3$ & $\mathcal{L}_4$ & $\mathcal{L}_5$ & $\mathcal{L}_6$ \\ \hline & & $x =$ & 1 1 0 0 & & & \\ & & $y =$ & 1 0 1 0 & & & \\ \hline $f_{0}$ & $f_{0000}$ && 0 0 0 0 & $(~)$ & $\operatorname{false}$ & $0$ \\ \hline $f_{1}$ & $f_{0001}$ && 0 0 0 1 & $(x)(y)$ & $\operatorname{neither}\ x\ \operatorname{nor}\ y$ & $\lnot x \land \lnot y$ \\ $f_{2}$ & $f_{0010}$ && 0 0 1 0 & $(x)\ y$ & $y\ \operatorname{without}\ x$ & $\lnot x \land y$ \\ $f_{4}$ & $f_{0100}$ && 0 1 0 0 & $x\ (y)$ & $x\ \operatorname{without}\ y$ & $x \land \lnot y$ \\ $f_{8}$ & $f_{1000}$ && 1 0 0 0 & $x\ y$ & $x\ \operatorname{and}\ y$ & $x \land y$ \\ \hline $f_{3}$ & $f_{0011}$ && 0 0 1 1 & $(x)$ & $\operatorname{not}\ x$ & $\lnot x$ \\ $f_{12}$ & $f_{1100}$ && 1 1 0 0 & $x$ & $x$ & $x$ \\ \hline $f_{6}$ & $f_{0110}$ && 0 1 1 0 & $(x,\ y)$ & $x\ \operatorname{not~equal~to}\ y$ & $x \ne y$ \\ $f_{9}$ & $f_{1001}$ && 1 0 0 1 & $((x,\ y))$ & $x\ \operatorname{equal~to}\ y$ & $x = y$ \\ \hline $f_{5}$ & $f_{0101}$ && 0 1 0 1 & $(y)$ & $\operatorname{not}\ y$ & $\lnot y$ \\ $f_{10}$ & $f_{1010}$ && 1 0 1 0 & $y$ & $y$ & $y$ \\ \hline $f_{7}$ & $f_{0111}$ && 0 1 1 1 & $(x\ y)$ & $\operatorname{not~both}\ x\ \operatorname{and}\ y$ & $\lnot x \lor \lnot y$ \\ $f_{11}$ & $f_{1011}$ && 1 0 1 1 & $(x\ (y))$ & $\operatorname{not}\ x\ \operatorname{without}\ y$ & $x \Rightarrow y$ \\ $f_{13}$ & $f_{1101}$ && 1 1 0 1 & $((x)\ y)$ & $\operatorname{not}\ y\ \operatorname{without}\ x$ & $x \Leftarrow y$ \\ $f_{14}$ & $f_{1110}$ && 1 1 1 0 & $((x)(y))$ & $x\ \operatorname{or}\ y$ & $x \lor y$ \\ \hline $f_{15}$ & $f_{1111}$ && 1 1 1 1 & $((~))$ & $\operatorname{true}$ & $1$ \\ \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}{\textbf{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 \end{tabular}\end{quote}

\subsection{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\textbf{Table A4. $\operatorname{D}f$ Expanded Over Differential Features $\{ \operatorname{d}x, \operatorname{d}y \}$}} \\ \hline & $f$ & $\operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y}$ & $\operatorname{D}f|_{\operatorname{d}x (\operatorname{d}y)}$ & $\operatorname{D}f|_{(\operatorname{d}x) \operatorname{d}y}$ & $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ \\ \hline $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline $f_{1}$ & $(x)(y)$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ $f_{2}$ & $(x)\ y$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ $f_{4}$ & $x\ (y)$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ $f_{8}$ & $x\ y$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ \hline $f_{3}$ & $(x)$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ $f_{12}$ & $x$ & $((~))$ & $((~))$ & $(~)$ & $(~)$ \\ \hline $f_{6}$ & $(x,\ y)$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ $f_{9}$ & $((x,\ y))$ & $(~)$ & $((~))$ & $((~))$ & $(~)$ \\ \hline $f_{5}$ & $(y)$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ $f_{10}$ & $y$ & $((~))$ & $(~)$ & $((~))$ & $(~)$ \\ \hline $f_{7}$ & $(x\ y)$ & $((x,\ y))$ & $y$ & $x$ & $(~)$ \\ $f_{11}$ & $(x\ (y))$ & $(x,\ y)$ & $(y)$ & $x$ & $(~)$ \\ $f_{13}$ & $((x)\ y)$ & $(x,\ y)$ & $y$ & $(x)$ & $(~)$ \\ $f_{14}$ & $((x)(y))$ & $((x,\ y))$ & $(y)$ & $(x)$ & $(~)$ \\ \hline $f_{15}$ & $((~))$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline \end{tabular}\end{quote}

\subsection{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\textbf{Table A5. $\operatorname{E}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\ \hline & $f$ & $\operatorname{E}f|_{x\ y}$ & $\operatorname{E}f|_{x (y)}$ & $\operatorname{E}f|_{(x) y}$ & $\operatorname{E}f|_{(x)(y)}$ \\ \hline $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline $f_{1}$ & $(x)(y)$ & $\operatorname{d}x\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $(\operatorname{d}x)(\operatorname{d}y)$ \\ $f_{2}$ & $(x)\ y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ & $(\operatorname{d}x)(\operatorname{d}y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ \\ $f_{4}$ & $x\ (y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $(\operatorname{d}x)(\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ \\ $f_{8}$ & $x\ y$ & $(\operatorname{d}x)(\operatorname{d}y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ \\ \hline $f_{3}$ & $(x)$ & $\operatorname{d}x$ & $\operatorname{d}x$ & $(\operatorname{d}x)$ & $(\operatorname{d}x)$ \\ $f_{12}$ & $x$ & $(\operatorname{d}x)$ & $(\operatorname{d}x)$ & $\operatorname{d}x$ & $\operatorname{d}x$ \\ \hline $f_{6}$ & $(x,\ y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $((\operatorname{d}x,\ \operatorname{d}y))$ & $((\operatorname{d}x,\ \operatorname{d}y))$ & $(\operatorname{d}x,\ \operatorname{d}y)$ \\ $f_{9}$ & $((x,\ y))$ & $((\operatorname{d}x,\ \operatorname{d}y))$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $((\operatorname{d}x,\ \operatorname{d}y))$ \\ \hline $f_{5}$ & $(y)$ & $\operatorname{d}y$ & $(\operatorname{d}y)$ & $\operatorname{d}y$ & $(\operatorname{d}y)$ \\ $f_{10}$ & $y$ & $(\operatorname{d}y)$ & $\operatorname{d}y$ & $(\operatorname{d}y)$ & $\operatorname{d}y$ \\ \hline $f_{7}$ & $(x\ y)$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $((\operatorname{d}x)\ \operatorname{d}y)$ & $(\operatorname{d}x\ (\operatorname{d}y))$ & $(\operatorname{d}x\ \operatorname{d}y)$ \\ $f_{11}$ & $(x\ (y))$ & $((\operatorname{d}x)\ \operatorname{d}y)$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $(\operatorname{d}x\ \operatorname{d}y)$ & $(\operatorname{d}x\ (\operatorname{d}y))$ \\ $f_{13}$ & $((x)\ y)$ & $(\operatorname{d}x\ (\operatorname{d}y))$ & $(\operatorname{d}x\ \operatorname{d}y)$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $((\operatorname{d}x)\ \operatorname{d}y)$ \\ $f_{14}$ & $((x)(y))$ & $(\operatorname{d}x\ \operatorname{d}y)$ & $(\operatorname{d}x\ (\operatorname{d}y))$ & $((\operatorname{d}x)\ \operatorname{d}y)$ & $((\operatorname{d}x)(\operatorname{d}y))$ \\ \hline $f_{15}$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ & $((~))$ \\ \hline \end{tabular}\end{quote}

\subsection{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}

\begin{quote}\begin{tabular}{|c|c||c|c|c|c|} \multicolumn{6}{c}{\textbf{Table A6. $\operatorname{D}f$ Expanded Over Ordinary Features $\{ x, y \}$}} \\ \hline & $f$ & $\operatorname{D}f|_{x\ y}$ & $\operatorname{D}f|_{x (y)}$ & $\operatorname{D}f|_{(x) y}$ & $\operatorname{D}f|_{(x)(y)}$ \\ \hline $f_{0}$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline $f_{1}$ & $(x)(y)$ & $\operatorname{d}x\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $((\operatorname{d}x)(\operatorname{d}y))$ \\ $f_{2}$ & $(x)\ y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $(\operatorname{d}x)\ \operatorname{d}y$ \\ $f_{4}$ & $x\ (y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $\operatorname{d}x\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ \\ $f_{8}$ & $x\ y$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ \\ \hline $f_{3}$ & $(x)$ & $\operatorname{d}x$ & $\operatorname{d}x$ & $\operatorname{d}x$ & $\operatorname{d}x$ \\ $f_{12}$ & $x$ & $\operatorname{d}x$ & $\operatorname{d}x$ & $\operatorname{d}x$ & $\operatorname{d}x$ \\ \hline $f_{6}$ & $(x,\ y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ \\ $f_{9}$ & $((x,\ y))$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ & $(\operatorname{d}x,\ \operatorname{d}y)$ \\ \hline $f_{5}$ & $(y)$ & $\operatorname{d}y$ & $\operatorname{d}y$ & $\operatorname{d}y$ & $\operatorname{d}y$ \\ $f_{10}$ & $y$ & $\operatorname{d}y$ & $\operatorname{d}y$ & $\operatorname{d}y$ & $\operatorname{d}y$ \\ \hline $f_{7}$ & $(x\ y)$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ \\ $f_{11}$ & $(x\ (y))$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $\operatorname{d}x\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ \\ $f_{13}$ & $((x)\ y)$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $\operatorname{d}x\ \operatorname{d}y$ & $((\operatorname{d}x)(\operatorname{d}y))$ & $(\operatorname{d}x)\ \operatorname{d}y$ \\ $f_{14}$ & $((x)(y))$ & $\operatorname{d}x\ \operatorname{d}y$ & $\operatorname{d}x\ (\operatorname{d}y)$ & $(\operatorname{d}x)\ \operatorname{d}y$ & $((\operatorname{d}x)(\operatorname{d}y))$ \\ \hline $f_{15}$ & $((~))$ & $(~)$ & $(~)$ & $(~)$ & $(~)$ \\ \hline \end{tabular}\end{quote}

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