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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 04:15, 11 November 2013

12,109 bytes added ,  04:15, 11 November 2013
update
Line 3,840: Line 3,840:  
& + & \texttt{(} u \texttt{)} v \cdot 0
 
& + & \texttt{(} u \texttt{)} v \cdot 0
 
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====
 +
 +
=====Computation of &epsilon;''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\boldsymbol\varepsilon f_{11}
 +
& = && f_{11}(u, v)
 +
\\[4pt]
 +
& = && \texttt{(} u \texttt{((} v \texttt{))}
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot f_{11}(1, 1)
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)
 +
\\[4pt]
 +
& = && \texttt{ } u \texttt{  } v \texttt{ }
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ }
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{11}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}\!</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of E''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{E}f_{11}
 +
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{(}
 +
\\
 +
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\\
 +
&&& \texttt{(}
 +
\\
 +
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\
 +
&&& \texttt{))}
 +
\\[4pt]
 +
& = &&
 +
u v
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[4pt]
 +
& = &&
 +
u v
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\!\cdot\!
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{E}f_{11}
 +
& = && \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}
 +
& + &  0
 +
\\[4pt]
 +
&& + & \texttt{ } u \texttt{  } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of D''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math>
 +
|
 +
<math>\begin{array}{*{10}{l}}
 +
\mathrm{D}f_{11}
 +
& = && \mathrm{E}f_{11}
 +
& + &  \boldsymbol\varepsilon f_{11}
 +
\\[4pt]
 +
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
 +
& + &  f_{11}(u, v)
 +
\\[4pt]
 +
& = &&
 +
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{(} v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = && 0
 +
& + &  0
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}
 +
& + &  0
 +
& + &  0
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
 +
& + &  0
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[4pt]
 +
&& + & 0
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
 +
& + &  \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  0
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + &  u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &  \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
 +
& + &  \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math>
 +
|
 +
<math>\begin{array}{c*{9}{l}}
 +
\mathrm{D}f_{11}
 +
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
 +
\\[20pt]
 +
\boldsymbol\varepsilon f_{11}
 +
& = & u v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{11}
 +
& = &
 +
u v
 +
\cdot
 +
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\cdot
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\cdot
 +
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\cdot
 +
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = &
 +
u v
 +
\cdot
 +
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}
 +
& + &
 +
u \texttt{(} v \texttt{)}
 +
\cdot
 +
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + &
 +
\texttt{(} u \texttt{)} v
 +
\cdot
 +
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}
 +
& + &
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\cdot
 +
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of d''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\Downarrow
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation of r''f''<sub>11</sub>=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
 +
\\[20pt]
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\\[20pt]
 +
\mathrm{r}f_{11}
 +
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
\end{array}</math>
 +
|}
 +
 +
<br>
 +
 +
=====Computation Summary for Implication=====
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
 +
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math>
 +
|
 +
<math>\begin{array}{c*{8}{l}}
 +
\boldsymbol\varepsilon f_{11}
 +
& = & u v \cdot 1
 +
& + & u \texttt{(} v \texttt{)} \cdot 0
 +
& + & \texttt{(} u \texttt{)} v \cdot 1
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
 +
\\[6pt]
 +
\mathrm{E}f_{11}
 +
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
 +
\\[6pt]
 +
\mathrm{D}f_{11}
 +
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
 +
\\[6pt]
 +
\mathrm{d}f_{11}
 +
& = & u v \cdot \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
 +
& + & 0
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
 +
\\[6pt]
 +
\mathrm{r}f_{11}
 +
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
 +
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
Line 3,906: Line 4,243:  
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
 
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)
 
\\[4pt]
 
\\[4pt]
−
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}
+
& = &&
 +
\texttt{((}
 +
\\
 +
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
 +
\\
 +
&&& \texttt{)(}
 +
\\
 +
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
 +
\\
 +
&&& \texttt{))}
 
\\[4pt]
 
\\[4pt]
 
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
 
& = && \texttt{ } u \texttt{  } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})
Jon Awbrey
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