# differential propositional calculus : appendix 4

## 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