differential propositional calculus : appendix 2

differential propositional calculus : appendix 2

\PMlinkescapephrase

action

\PMlinkescapephrase

Action

\PMlinkescapephrase

actions

\PMlinkescapephrase

Actions

\PMlinkescapephrase

algebraic

\PMlinkescapephrase

Algebraic

\PMlinkescapephrase

basis

\PMlinkescapephrase

Basis

\PMlinkescapephrase

expanded

\PMlinkescapephrase

Expanded

\PMlinkescapephrase

expands

\PMlinkescapephrase

Expands

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 forms

over a

logical basis

:

{

(

d

⁡

x

)

⁢

(

d

⁡

y

)

,

d

⁡

x

⁢

(

d

⁡

y

)

,

(

d

⁡

x

)

⁢

d

⁡

y

,

d

⁡

x

⁢

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:

∂

⁡

x

=

d

⁡

x

⁢

(

d

⁡

y

)

and

∂

⁡

y

=

(

d

⁡

x

)

⁢

d

⁡

y

.

Table A8 expands the resulting differential forms over an

algebraic basis

:

{

1

,

d

⁡

x

,

d

⁡

y

,

d

⁡

x

⁢

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:

0.1

Table A7. Differential Forms Expanded on a Logical Basis

0.2

Table A8. Differential Forms Expanded on an Algebraic Basis

0.1

Table A7. Differential Forms Expanded on a Logical Basis

Table A7. Differential Forms Expanded on a Logical Basis

f

D

⁡

f

d

⁡

f

f

0

(

)

0

0

f

1

f

2

f

4

f

8

(

x

)

(

y

)

(

x

)

y

x

(

y

)

x

y

(

y

)

d

⁡

x

⁢

(

d

⁡

y

)

+

(

x

)

(

d

⁡

x

)

⁢

d

⁡

y

+

(

(

x

,

y

)

)

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

⁢

(

d

⁡

y

)

+

(

x

)

(

d

⁡

x

)

⁢

d

⁡

y

+

(

x

,

y

)

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

⁢

(

d

⁡

y

)

+

x

(

d

⁡

x

)

⁢

d

⁡

y

+

(

x

,

y

)

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

⁢

(

d

⁡

y

)

+

x

(

d

⁡

x

)

⁢

d

⁡

y

+

(

(

x

,

y

)

)

d

⁡

x

⁢

d

⁡

y

(

y

)

∂

⁡

x

+

(

x

)

∂

⁡

y

y

∂

⁡

x

+

(

x

)

∂

⁡

y

(

y

)

∂

⁡

x

+

x

∂

⁡

y

y

∂

⁡

x

+

x

∂

⁡

y

f

3

f

12

(

x

)

x

d

⁡

x

⁢

(

d

⁡

y

)

+

d

⁡

x

⁢

d

⁡

y

d

⁡

x

⁢

(

d

⁡

y

)

+

d

⁡

x

⁢

d

⁡

y

∂

⁡

x

∂

⁡

x

f

6

f

9

(

x

,

y

)

(

(

x

,

y

)

)

d

⁡

x

⁢

(

d

⁡

y

)

+

(

d

⁡

x

)

⁢

d

⁡

y

d

⁡

x

⁢

(

d

⁡

y

)

+

(

d

⁡

x

)

⁢

d

⁡

y

∂

⁡

x

+

∂

⁡

y

∂

⁡

x

+

∂

⁡

y

f

5

f

10

(

y

)

y

(

d

⁡

x

)

⁢

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

(

d

⁡

x

)

⁢

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

∂

⁡

y

∂

⁡

y

f

7

f

11

f

13

f

14

(

x

y

)

(

x

(

y

)

)

(

(

x

)

y

)

(

(

x

)

(

y

)

)

y

d

⁡

x

⁢

(

d

⁡

y

)

+

x

(

d

⁡

x

)

⁢

d

⁡

y

+

(

(

x

,

y

)

)

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

⁢

(

d

⁡

y

)

+

x

(

d

⁡

x

)

⁢

d

⁡

y

+

(

x

,

y

)

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

⁢

(

d

⁡

y

)

+

(

x

)

(

d

⁡

x

)

⁢

d

⁡

y

+

(

x

,

y

)

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

⁢

(

d

⁡

y

)

+

(

x

)

(

d

⁡

x

)

⁢

d

⁡

y

+

(

(

x

,

y

)

)

d

⁡

x

⁢

d

⁡

y

y

∂

⁡

x

+

x

∂

⁡

y

(

y

)

∂

⁡

x

+

x

∂

⁡

y

y

∂

⁡

x

+

(

x

)

∂

⁡

y

(

y

)

∂

⁡

x

+

(

x

)

∂

⁡

y

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

D

⁡

f

d

⁡

f

f

0

(

)

0

0

f

1

f

2

f

4

f

8

(

x

)

(

y

)

(

x

)

y

x

(

y

)

x

y

(

y

)

d

⁡

x

+

(

x

)

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

+

(

x

)

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

+

x

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

+

x

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

+

(

x

)

d

⁡

y

y

d

⁡

x

+

(

x

)

d

⁡

y

(

y

)

d

⁡

x

+

x

d

⁡

y

y

d

⁡

x

+

x

d

⁡

y

f

3

f

12

(

x

)

x

d

⁡

x

d

⁡

x

d

⁡

x

d

⁡

x

f

6

f

9

(

x

,

y

)

(

(

x

,

y

)

)

d

⁡

x

+

d

⁡

y

d

⁡

x

+

d

⁡

y

d

⁡

x

+

d

⁡

y

d

⁡

x

+

d

⁡

y

f

5

f

10

(

y

)

y

d

⁡

y

d

⁡

y

d

⁡

y

d

⁡

y

f

7

f

11

f

13

f

14

(

x

y

)

(

x

(

y

)

)

(

(

x

)

y

)

(

(

x

)

(

y

)

)

y

d

⁡

x

+

x

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

+

x

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

+

(

x

)

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

(

y

)

d

⁡

x

+

(

x

)

d

⁡

y

+

d

⁡

x

⁢

d

⁡

y

y

d

⁡

x

+

x

d

⁡

y

(

y

)

d

⁡

x

+

x

d

⁡

y

y

d

⁡

x

+

(

x

)

d

⁡

y

(

y

)

d

⁡

x

+

(

x

)

d

⁡

y

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

Generated on Fri Feb 9 22:24:54 2018 by

LaTeXML