differential propositional calculus : appendix 4


Contents

Detail of Calculation for the Difference Map

Detail of Calculation for $ \operatorname{D}f = \operatorname{E}f + f$
  \begin{displaymath}\begin{array}{cr}
& \operatorname{E}f\vert _{\operatorname{d}...
...}f\vert _{\operatorname{d}x\ \operatorname{d}y} \\
\end{array}\end{displaymath} \begin{displaymath}\begin{array}{cr}
& \operatorname{E}f\vert _{\operatorname{d}...
...\vert _{\operatorname{d}x\ (\operatorname{d}y)} \\
\end{array}\end{displaymath} \begin{displaymath}\begin{array}{cr}
& \operatorname{E}f\vert _{(\operatorname{d...
...\vert _{(\operatorname{d}x)\ \operatorname{d}y} \\
\end{array}\end{displaymath} \begin{displaymath}\begin{array}{cr}
& \operatorname{E}f\vert _{(\operatorname{d...
...\vert _{(\operatorname{d}x)(\operatorname{d}y)} \\
\end{array}\end{displaymath}
$ f_{0}$ $ 0 + 0 = 0$ $ 0 + 0 = 0$ $ 0 + 0 = 0$ $ 0 + 0 = 0$
$ f_{1}$ $ \begin{smallmatrix}
& x\ y & \operatorname{d}x & \operatorname{d}y \\
+ & (x)...
... \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x\ (y) & \operatorname{d}x & (\operatorname{d}y) \\
+ &...
...}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x)\ y & (\operatorname{d}x) & \operatorname{d}y \\
+ &...
...d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x)(y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{2}$ $ \begin{smallmatrix}
& x\ (y) & \operatorname{d}x & \operatorname{d}y \\
+ & (...
...}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x\ y & \operatorname{d}x & (\operatorname{d}y) \\
+ & (...
...{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x) (y) & (\operatorname{d}x) & \operatorname{d}y \\
+ ...
...d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
+...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{4}$ $ \begin{smallmatrix}
& (x)\ y & \operatorname{d}x & \operatorname{d}y \\
+ & x...
...}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x) (y) & \operatorname{d}x & (\operatorname{d}y) \\
+ ...
...}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x\ y & (\operatorname{d}x) & \operatorname{d}y \\
+ & x...
...e{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{8}$ $ \begin{smallmatrix}
& (x)(y) & \operatorname{d}x & \operatorname{d}y \\
+ & x...
... \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x)\ y & \operatorname{d}x & (\operatorname{d}y) \\
+ &...
...{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x\ (y) & (\operatorname{d}x) & \operatorname{d}y \\
+ &...
...e{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
+ &...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{3}$ $ \begin{smallmatrix}
& x & \operatorname{d}x & \operatorname{d}y \\
+ & (x) & ...
...ame{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x & \operatorname{d}x & (\operatorname{d}y) \\
+ & (x) ...
...{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x) & (\operatorname{d}x) & \operatorname{d}y \\
+ & (x...
...e{d}y \\
= & 0 & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{12}$ $ \begin{smallmatrix}
& (x) & \operatorname{d}x & \operatorname{d}y \\
+ & x & ...
...ame{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x) & \operatorname{d}x & (\operatorname{d}y) \\
+ & x ...
...{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x & (\operatorname{d}x) & \operatorname{d}y \\
+ & x & ...
...e{d}y \\
= & 0 & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& x & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & x ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{6}$ $ \begin{smallmatrix}
& (x, y) & \operatorname{d}x & \operatorname{d}y \\
+ & (...
...ame{d}y \\
= & 0 & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x, y)) & \operatorname{d}x & (\operatorname{d}y) \\
+...
...{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x, y)) & (\operatorname{d}x) & \operatorname{d}y \\
+...
...e{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x, y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{9}$ $ \begin{smallmatrix}
& ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
+ &...
...ame{d}y \\
= & 0 & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x, y) & \operatorname{d}x & (\operatorname{d}y) \\
+ &...
...{d}y) \\
= & 1 & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x, y) & (\operatorname{d}x) & \operatorname{d}y \\
+ &...
...e{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x, y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{5}$ $ \begin{smallmatrix}
& y & \operatorname{d}x & \operatorname{d}y \\
+ & (y) & ...
...ame{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (y) & \operatorname{d}x & (\operatorname{d}y) \\
+ & (y...
...{d}y) \\
= & 0 & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& y & (\operatorname{d}x) & \operatorname{d}y \\
+ & (y) ...
...e{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{10}$ $ \begin{smallmatrix}
& (y) & \operatorname{d}x & \operatorname{d}y \\
+ & y & ...
...ame{d}y \\
= & 1 & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& y & \operatorname{d}x & (\operatorname{d}y) \\
+ & y & ...
...{d}y) \\
= & 0 & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (y) & (\operatorname{d}x) & \operatorname{d}y \\
+ & y ...
...e{d}y \\
= & 1 & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& y & (\operatorname{d}x) & (\operatorname{d}y) \\
+ & y ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{7}$ $ \begin{smallmatrix}
& ((x)(y)) & \operatorname{d}x & \operatorname{d}y \\
+ &...
... \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x)\ y) & \operatorname{d}x & (\operatorname{d}y) \\
+...
...{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x\ (y)) & (\operatorname{d}x) & \operatorname{d}y \\
+...
...e{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
+...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{11}$ $ \begin{smallmatrix}
& ((x)\ y) & \operatorname{d}x & \operatorname{d}y \\
+ &...
...}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x) (y)) & \operatorname{d}x & (\operatorname{d}y) \\
...
...}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x\ y) & (\operatorname{d}x) & \operatorname{d}y \\
+ &...
...e{d}y \\
= & x & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{13}$ $ \begin{smallmatrix}
& (x\ (y)) & \operatorname{d}x & \operatorname{d}y \\
+ &...
...}y \\
= & (x, y) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x\ y) & \operatorname{d}x & (\operatorname{d}y) \\
+ &...
...{d}y) \\
= & y & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x) (y)) & (\operatorname{d}x) & \operatorname{d}y \\
...
...d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\ ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{14}$ $ \begin{smallmatrix}
& (x\ y) & \operatorname{d}x & \operatorname{d}y \\
+ & (...
... \\
= & ((x, y)) & \operatorname{d}x & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& (x\ (y)) & \operatorname{d}x & (\operatorname{d}y) \\
+...
...}y) \\
= & (y) & \operatorname{d}x & (\operatorname{d}y) \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x)\ y) & (\operatorname{d}x) & \operatorname{d}y \\
+...
...d}y \\
= & (x) & (\operatorname{d}x) & \operatorname{d}y \\
\end{smallmatrix}$ $ \begin{smallmatrix}
& ((x)(y)) & (\operatorname{d}x) & (\operatorname{d}y) \\ ...
...}y) \\
= & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
\end{smallmatrix}$
$ f_{15}$ $ 1 + 1 = 0$ $ 1 + 1 = 0$ $ 1 + 1 = 0$ $ 1 + 1 = 0$



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As of this snapshot date, this entry was owned by Jon Awbrey.