12,089
edits
Changes
Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 17:44, 27 May 2008
, 17:44, 27 May 2008→Differential Propositions: HTML -- TeX
|+ '''Table 1. Propositional Forms on Two Variables'''
|+ '''Table 1. Propositional Forms on Two Variables'''
|- style="background:paleturquoise"
|- style="background:paleturquoise"
! style="width:15%" | L<sub>1</sub>
! style="width:15%" | <math>\mathcal{L}_1</math>
! style="width:15%" | L<sub>2</sub>
! style="width:15%" | <math>\mathcal{L}_2</math>
! style="width:15%" | L<sub>3</sub>
! style="width:15%" | <math>\mathcal{L}_3</math>
! style="width:15%" | L<sub>4</sub>
! style="width:15%" | <math>\mathcal{L}_4</math>
! style="width:15%" | L<sub>5</sub>
! style="width:15%" | <math>\mathcal{L}_5</math>
! style="width:15%" | L<sub>6</sub>
! style="width:15%" | <math>\mathcal{L}_6</math>
|- style="background:paleturquoise"
|- style="background:paleturquoise"
|
|
| align="right" | x :
| align="right" | <math>x\!</math> :
| 1 1 0 0
| 1 1 0 0
|
|
|- style="background:paleturquoise"
|- style="background:paleturquoise"
|
|
| align="right" | y :
| align="right" | <math>y\!</math> :
| 1 0 1 0
| 1 0 1 0
|
|
|
|
|-
|-
| f<sub>0</sub>
| <math>f_{0}\!</math>
| f<sub>0000</sub>
| <math>f_{0000}\!</math>
| 0 0 0 0
| 0 0 0 0
| ( )
| ( )
| 0
| 0
|-
|-
| f<sub>1</sub>
| <math>f_{1}\!</math>
| f<sub>0001</sub>
| <math>f_{0001}\!</math>
| 0 0 0 1
| 0 0 0 1
| (x)(y)
| (x)(y)
| ¬x ∧ ¬y
| ¬x ∧ ¬y
|-
|-
| f<sub>2</sub>
| <math>f_{2}\!</math>
| f<sub>0010</sub>
| <math>f_{0010}\!</math>
| 0 0 1 0
| 0 0 1 0
| (x) y
| (x) y
| ¬x ∧ y
| ¬x ∧ y
|-
|-
| f<sub>3</sub>
| <math>f_{3}\!</math>
| f<sub>0011</sub>
| <math>f_{0011}\!</math>
| 0 0 1 1
| 0 0 1 1
| (x)
| (x)
| ¬x
| ¬x
|-
|-
| f<sub>4</sub>
| <math>f_{4}\!</math>
| f<sub>0100</sub>
| <math>f_{0100}\!</math>
| 0 1 0 0
| 0 1 0 0
| x (y)
| x (y)
| x ∧ ¬y
| x ∧ ¬y
|-
|-
| f<sub>5</sub>
| <math>f_{5}\!</math>
| f<sub>0101</sub>
| <math>f_{0101}\!</math>
| 0 1 0 1
| 0 1 0 1
| (y)
| (y)
| ¬y
| ¬y
|-
|-
| f<sub>6</sub>
| <math>f_{6}\!</math>
| f<sub>0110</sub>
| <math>f_{0110}\!</math>
| 0 1 1 0
| 0 1 1 0
| (x, y)
| (x, y)
| x ≠ y
| x ≠ y
|-
|-
| f<sub>7</sub>
| <math>f_{7}\!</math>
| f<sub>0111</sub>
| <math>f_{0111}\!</math>
| 0 1 1 1
| 0 1 1 1
| (x y)
| (x y)
| ¬x ∨ ¬y
| ¬x ∨ ¬y
|-
|-
| f<sub>8</sub>
| <math>f_{8}\!</math>
| f<sub>1000</sub>
| <math>f_{1000}\!</math>
| 1 0 0 0
| 1 0 0 0
| x y
| x y
| x ∧ y
| x ∧ y
|-
|-
| f<sub>9</sub>
| <math>f_{9}\!</math>
| f<sub>1001</sub>
| <math>f_{1001}\!</math>
| 1 0 0 1
| 1 0 0 1
| ((x, y))
| ((x, y))
| x = y
| x = y
|-
|-
| f<sub>10</sub>
| <math>f_{10}\!</math>
| f<sub>1010</sub>
| <math>f_{1010}\!</math>
| 1 0 1 0
| 1 0 1 0
| y
| y
| y
| y
|-
|-
| f<sub>11</sub>
| <math>f_{11}\!</math>
| f<sub>1011</sub>
| <math>f_{1011}\!</math>
| 1 0 1 1
| 1 0 1 1
| (x (y))
| (x (y))
| x → y
| x → y
|-
|-
| f<sub>12</sub>
| <math>f_{12}\!</math>
| f<sub>1100</sub>
| <math>f_{1100}\!</math>
| 1 1 0 0
| 1 1 0 0
| x
| x
| x
| x
|-
|-
| f<sub>13</sub>
| <math>f_{13}\!</math>
| f<sub>1101</sub>
| <math>f_{1101}\!</math>
| 1 1 0 1
| 1 1 0 1
| ((x) y)
| ((x) y)
| x ← y
| x ← y
|-
|-
| f<sub>14</sub>
| <math>f_{14}\!</math>
| f<sub>1110</sub>
| <math>f_{1110}\!</math>
| 1 1 1 0
| 1 1 1 0
| ((x)(y))
| ((x)(y))
| x ∨ y
| x ∨ y
|-
|-
| f<sub>15</sub>
| <math>f_{15}\!</math>
| f<sub>1111</sub>
| <math>f_{1111}\!</math>
| 1 1 1 1
| 1 1 1 1
| (( ))
| (( ))
<br>
<br>
The next four Tables expand the expressions of ''Ef'' and ''Df'' in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
<pre>
<pre>