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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 18:08, 8 June 2008

6,152 bytes added ,  18:08, 8 June 2008
→‎Appendices @ PlanetMath : TeX Format: update
Line 435: Line 435:  
$f_{0}$  & $f_{0000}$ & & 0 0 0 0 & $(~)$      & false                & $0$ \\
 
$f_{0}$  & $f_{0000}$ & & 0 0 0 0 & $(~)$      & false                & $0$ \\
 
$f_{1}$  & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$    & neither $x$ nor $y$  & $\lnot x \land \lnot y  $ \\
 
$f_{1}$  & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$    & neither $x$ nor $y$  & $\lnot x \land \lnot y  $ \\
−
$f_{2}$  & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$    & $y$ and not $x$      & $\lnot x \land y$ \\
+
$f_{2}$  & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$    & $y$ without $x$      & $\lnot x \land y$ \\
 
$f_{3}$  & $f_{0011}$ & & 0 0 1 1 & $(x)$      & not $x$              & $\lnot x$ \\
 
$f_{3}$  & $f_{0011}$ & & 0 0 1 1 & $(x)$      & not $x$              & $\lnot x$ \\
−
$f_{4}$  & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$    & $x$ and not $y$      & $x \land \lnot y$ \\
+
$f_{4}$  & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$    & $x$ without $y$      & $x \land \lnot y$ \\
 
$f_{5}$  & $f_{0101}$ & & 0 1 0 1 & $(y)$      & not $y$              & $\lnot y$ \\
 
$f_{5}$  & $f_{0101}$ & & 0 1 0 1 & $(y)$      & not $y$              & $\lnot y$ \\
 
$f_{6}$  & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$  & $x$ not equal to $y$ & $x \ne y$ \\
 
$f_{6}$  & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$  & $x$ not equal to $y$ & $x \ne y$ \\
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\hline
 
\hline
 
$f_{1}$  & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$    & neither $x$ nor $y$  & $\lnot x \land \lnot y  $ \\
 
$f_{1}$  & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$    & neither $x$ nor $y$  & $\lnot x \land \lnot y  $ \\
−
$f_{2}$  & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$    & $y$ and not $x$      & $\lnot x \land y$ \\
+
$f_{2}$  & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$    & $y$ without $x$      & $\lnot x \land y$ \\
−
$f_{4}$  & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$    & $x$ and not $y$      & $x \land \lnot y$ \\
+
$f_{4}$  & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$    & $x$ without $y$      & $x \land \lnot y$ \\
 
$f_{8}$  & $f_{1000}$ & & 1 0 0 0 & $x\ y$      & $x$ and $y$          & $x \land y$ \\
 
$f_{8}$  & $f_{1000}$ & & 1 0 0 0 & $x\ y$      & $x$ and $y$          & $x \land y$ \\
 
\hline
 
\hline
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\hline
 
\hline
 
$f_{15}$ & $(~)$      & $(~)$      & $(~)$  & $(~)$  & $(~)$ \\
 
$f_{15}$ & $(~)$      & $(~)$      & $(~)$  & $(~)$  & $(~)$ \\
 +
\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}{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}{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
 
\hline
 
\end{tabular}\end{quote}
 
\end{tabular}\end{quote}
Jon Awbrey
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