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differential propositional calculus : appendix 4

Contents:

0.1 Detail of Calculation for the Difference Map

0.1 Detail of Calculation for the Difference Map

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

Title	differential propositional calculus : appendix 4
Canonical name	DifferentialPropositionalCalculusAppendix4
Date of creation	2013-03-22 18:09:25
Last modified on	2013-03-22 18:09:25
Owner	Jon Awbrey (15246)
Last modified by	Jon Awbrey (15246)
Numerical id	7
Author	Jon Awbrey (15246)
Entry type	Application
Classification	msc 53A40
Classification	msc 39A12
Classification	msc 34G99
Classification	msc 03B44
Classification	msc 03B05
Classification	msc 03B42
Related topic	DifferentialLogic
Related topic	MinimalNegationOperator
Related topic	PropositionalCalculus
Related topic	ZerothOrderLogic