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[parent] differential propositional calculus : appendix 2 (Application)

The actions of the difference operator $\operatorname{D}$ and the tangent operator $\operatorname{d}$ 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:

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

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:

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

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

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

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.


Contents

Table A7. Differential Forms Expanded on a Logical Basis

Table A7. Differential Forms Expanded on a Logical Basis
  $f$ $\operatorname{D}f$ $\operatorname{d}f$
$f_{0}$ $(~)$ 0 0
$\begin{smallmatrix} f_{1} \ f_{2} \ f_{4} \ f_{8} \ \end{smallmatrix}$ $\begin{smallmatrix} (x) & (y) \ (x) & y \ x & (y) \ x & y \ \end{smallmatrix}$ $\begin{smallmatrix} (y) & \operatorname{d}x\ (\operatorname{d}y) & + & (x) & (\... ...d}y & + & ((x, y)) & \operatorname{d}x\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} (y) & \partial x & + & (x) & \partial y \ y & \partial x ... ... & x & \partial y \ y & \partial x & + & x & \partial y \ \end{smallmatrix}$
$\begin{smallmatrix} f_{3} \ f_{12} \ \end{smallmatrix}$ $\begin{smallmatrix} (x) \ x \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}x\ (\operatorname{d}y) & + & \operatorname{... ...ratorname{d}y) & + & \operatorname{d}x\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} \partial x \ \partial x \ \end{smallmatrix}$
$\begin{smallmatrix} f_{6} \ f_{9} \ \end{smallmatrix}$ $\begin{smallmatrix} (x, & y) \ ((x, & y)) \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}x\ (\operatorname{d}y) & + & (\operatorname... ...torname{d}y) & + & (\operatorname{d}x)\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} \partial x & + & \partial y \ \partial x & + & \partial y \ \end{smallmatrix}$
$\begin{smallmatrix} f_{5} \ f_{10} \ \end{smallmatrix}$ $\begin{smallmatrix} (y) \ y \ \end{smallmatrix}$ $\begin{smallmatrix} (\operatorname{d}x)\ \operatorname{d}y & + & \operatorname{... ...eratorname{d}y & + & \operatorname{d}x\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} \partial y \ \partial y \ \end{smallmatrix}$
$\begin{smallmatrix} f_{7} \ f_{11} \ f_{13} \ f_{14} \ \end{smallmatrix}$ $\begin{smallmatrix} (x & y) \ (x & (y)) \ ((x) & y) \ ((x) & (y)) \ \end{smallmatrix}$ $\begin{smallmatrix} y & \operatorname{d}x\ (\operatorname{d}y) & + & x & (\oper... ...d}y & + & ((x, y)) & \operatorname{d}x\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} y & \partial x & + & x & \partial y \ (y) & \partial x & ... ... & \partial y \ (y) & \partial x & + & (x) & \partial y \ \end{smallmatrix}$
$f_{15}$ $((~))$ 0 0

Table A8. Differential Forms Expanded on an Algebraic Basis

Table A8. Differential Forms Expanded on an Algebraic Basis
  $f$ $\operatorname{D}f$ $\operatorname{d}f$
$f_{0}$ $(~)$ 0 0
$\begin{smallmatrix} f_{1} \ f_{2} \ f_{4} \ f_{8} \ \end{smallmatrix}$ $\begin{smallmatrix} (x) & (y) \ (x) & y \ x & (y) \ x & y \ \end{smallmatrix}$ $\begin{smallmatrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y & + &... ...eratorname{d}y & + & \operatorname{d}x\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \ y... ...d}y \ y & \operatorname{d}x & + & x & \operatorname{d}y \ \end{smallmatrix}$
$\begin{smallmatrix} f_{3} \ f_{12} \ \end{smallmatrix}$ $\begin{smallmatrix} (x) \ x \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}x \ \operatorname{d}x \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}x \ \operatorname{d}x \ \end{smallmatrix}$
$\begin{smallmatrix} f_{6} \ f_{9} \ \end{smallmatrix}$ $\begin{smallmatrix} (x, & y) \ ((x, & y)) \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}x & + & \operatorname{d}y \ \operatorname{d}x & + & \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}x & + & \operatorname{d}y \ \operatorname{d}x & + & \operatorname{d}y \ \end{smallmatrix}$
$\begin{smallmatrix} f_{5} \ f_{10} \ \end{smallmatrix}$ $\begin{smallmatrix} (y) \ y \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}y \ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} \operatorname{d}y \ \operatorname{d}y \ \end{smallmatrix}$
$\begin{smallmatrix} f_{7} \ f_{11} \ f_{13} \ f_{14} \ \end{smallmatrix}$ $\begin{smallmatrix} (x & y) \ (x & (y)) \ ((x) & y) \ ((x) & (y)) \ \end{smallmatrix}$ $\begin{smallmatrix} y & \operatorname{d}x & + & x & \operatorname{d}y & + & \op... ...eratorname{d}y & + & \operatorname{d}x\ \operatorname{d}y \ \end{smallmatrix}$ $\begin{smallmatrix} y & \operatorname{d}x & + & x & \operatorname{d}y \ (y) &... ...\ (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \ \end{smallmatrix}$
$f_{15}$ $((~))$ 0 0



"differential propositional calculus : appendix 2" is owned by Jon Awbrey.

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See Also: differential logic, minimal negation operator


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Cross-references: positive propositions, differential basis, differential propositions, tangent universe, differential variables, singular propositions

This is version 1 of differential propositional calculus : appendix 2, born on 2009-05-25.
Object id is 779, canonical name is DifferentialPropositionalCalculusAppendix2.
Accessed 348 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 02.10.Ab (Logic and set theory)
 02.10.Ox (Combinatorics; graph theory)
 02.10.Ud (Linear algebra)
 02.20.-a (Group theory )
 02.30.-f (Function theory, analysis)
 02.40.-k (Geometry, differential geometry, and topology )
 02.40.Yy (Geometric mechanics )
 02.50.Tt (Inference methods)
 02.70.-c (Computational techniques )
 02.70.Bf (Finite-difference methods)
 02.70.Wz (Symbolic computation )
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