differential propositional calculus : appendix 2

action Action actions Actions algebraic Algebraic basis Basis expanded Expanded expands Expands

The actions of the difference operator (http://planetmath.org/FiniteDifference) $\mathrm{D}$ and the tangent operator (http://planetmath.org/TangentMap) $\mathrm{d}$ on the 16 propositional forms in two variables are shown in the Tables below.

Table A7 expands the resulting differential forms over a logical basis:

$\{(\mathrm{d}x)(\mathrm{d}y),\mathrm{d}x(\mathrm{d}y),(\mathrm{d}x)\mathrm{d}y,\mathrm{d}x\mathrm{d}y\}.$

This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive cells of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:

$\partial x=\mathrm{d}x(\mathrm{d}y)$ and $\partial y=(\mathrm{d}x)\mathrm{d}y.$

Table A8 expands the resulting differential forms over an algebraic basis:

$\{1,\mathrm{d}x,\mathrm{d}y,\mathrm{d}x\mathrm{d}y\}.$

This set consists of the positive propositions in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the positive differential basis.

0.1 Table A7. Differential Forms Expanded on a Logical Basis

Table A7. Differential Forms Expanded on a Logical Basis
	$f$	$\mathrm{D}f$	$\mathrm{d}f$
$f_0$	$()$	$0$	$0$
$\begin{array}{c}\hfill f_1\hfill \\ \hfill f_2\hfill \\ \hfill f_4\hfill \\ \hfill f_8\hfill \end{array}$
$\begin{array}{c}\hfill f_3\hfill \\ \hfill f_{12}\hfill \end{array}$	$\begin{array}{c}\hfill \left(x\right)\hfill \\ \hfill x\hfill \end{array}$		$\begin{array}{c}\hfill \partial x\hfill \\ \hfill \partial x\hfill \end{array}$
$\begin{array}{c}\hfill f_6\hfill \\ \hfill f_9\hfill \end{array}$
$\begin{array}{c}\hfill f_5\hfill \\ \hfill f_{10}\hfill \end{array}$	$\begin{array}{c}\hfill \left(y\right)\hfill \\ \hfill y\hfill \end{array}$		$\begin{array}{c}\hfill \partial y\hfill \\ \hfill \partial y\hfill \end{array}$
$\begin{array}{c}\hfill f_7\hfill \\ \hfill f_{11}\hfill \\ \hfill f_{13}\hfill \\ \hfill f_{14}\hfill \end{array}$
$f_{15}$	$(())$	$0$	$0$

0.2 Table A8. Differential Forms Expanded on an Algebraic Basis

Table A8. Differential Forms Expanded on an Algebraic Basis
	$f$	$\mathrm{D}f$	$\mathrm{d}f$
$f_0$	$()$	$0$	$0$
$\begin{array}{c}\hfill f_1\hfill \\ \hfill f_2\hfill \\ \hfill f_4\hfill \\ \hfill f_8\hfill \end{array}$
$\begin{array}{c}\hfill f_3\hfill \\ \hfill f_{12}\hfill \end{array}$	$\begin{array}{c}\hfill \left(x\right)\hfill \\ \hfill x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}x\hfill \\ \hfill \mathrm{d}x\hfill \end{array}$
$\begin{array}{c}\hfill f_6\hfill \\ \hfill f_9\hfill \end{array}$
$\begin{array}{c}\hfill f_5\hfill \\ \hfill f_{10}\hfill \end{array}$	$\begin{array}{c}\hfill \left(y\right)\hfill \\ \hfill y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$	$\begin{array}{c}\hfill \mathrm{d}y\hfill \\ \hfill \mathrm{d}y\hfill \end{array}$
$\begin{array}{c}\hfill f_7\hfill \\ \hfill f_{11}\hfill \\ \hfill f_{13}\hfill \\ \hfill f_{14}\hfill \end{array}$
$f_{15}$	$(())$	$0$	$0$

Title	differential propositional calculus : appendix 2
Canonical name	DifferentialPropositionalCalculusAppendix2
Date of creation	2013-03-22 18:09:13
Last modified on	2013-03-22 18:09:13
Owner	Jon Awbrey (15246)
Last modified by	Jon Awbrey (15246)
Numerical id	15
Author	Jon Awbrey (15246)
Entry type	Application
Classification	msc 53A40
Classification	msc 39A12
Classification	msc 34G99
Classification	msc 03B44
Classification	msc 03B42
Classification	msc 03B05
Related topic	DifferentialLogic
Related topic	MinimalNegationOperator
Related topic	PropositionalCalculus
Related topic	ZerothOrderLogic