differential propositional calculus : appendix 2


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

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

{(dx)(dy),dx(dy),(dx)dy,dxdy}.

This set consists of the singular propositionsPlanetmathPlanetmathPlanetmath 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:

x=dx(dy) and y=(dx)dy.

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

{1,dx,dy,dxdy}.

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 Df df
f0 () 0 0
f1f2f4f8 (x)(y)(x)yx(y)xy (y)dx(dy)+(x)(dx)dy+((x,y))dxdyydx(dy)+(x)(dx)dy+(x,y)dxdy(y)dx(dy)+x(dx)dy+(x,y)dxdyydx(dy)+x(dx)dy+((x,y))dxdy (y)x+(x)yyx+(x)y(y)x+xyyx+xy
f3f12 (x)x dx(dy)+dxdydx(dy)+dxdy xx
f6f9 (x,y)((x,y)) dx(dy)+(dx)dydx(dy)+(dx)dy x+yx+y
f5f10 (y)y (dx)dy+dxdy(dx)dy+dxdy yy
f7f11f13f14 (xy)(x(y))((x)y)((x)(y)) ydx(dy)+x(dx)dy+((x,y))dxdy(y)dx(dy)+x(dx)dy+(x,y)dxdyydx(dy)+(x)(dx)dy+(x,y)dxdy(y)dx(dy)+(x)(dx)dy+((x,y))dxdy yx+xy(y)x+xyyx+(x)y(y)x+(x)y
f15 (()) 0 0

0.2 Table A8. Differential Forms Expanded on an Algebraic Basis

Table A8. Differential Forms Expanded on an Algebraic Basis
f Df df
f0 () 0 0
f1f2f4f8 (x)(y)(x)yx(y)xy (y)dx+(x)dy+dxdyydx+(x)dy+dxdy(y)dx+xdy+dxdyydx+xdy+dxdy (y)dx+(x)dyydx+(x)dy(y)dx+xdyydx+xdy
f3f12 (x)x dxdx dxdx
f6f9 (x,y)((x,y)) dx+dydx+dy dx+dydx+dy
f5f10 (y)y dydy dydy
f7f11f13f14 (xy)(x(y))((x)y)((x)(y)) ydx+xdy+dxdy(y)dx+xdy+dxdyydx+(x)dy+dxdy(y)dx+(x)dy+dxdy ydx+xdy(y)dx+xdyydx+(x)dy(y)dx+(x)dy
f15 (()) 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