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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 22:08, 11 September 2013

179 bytes added ,  22:08, 11 September 2013
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
 
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math>
 
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math>
 
|- style="height:40px; background:ghostwhite; width:100%"
 
|- style="height:40px; background:ghostwhite; width:100%"
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| style="width:10%" | <math>u\!</math>
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| style="width:8%" | <math>u\!</math>
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| style="width:10%" | <math>v\!</math>
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| style="width:8%" | <math>v\!</math>
 
| style="width:12%; border-left:1px solid black" | <math>x\!</math>
 
| style="width:12%; border-left:1px solid black" | <math>x\!</math>
 
| style="width:12%" | <math>y\!</math>
 
| style="width:12%" | <math>y\!</math>
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| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math>
 
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math>
 
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
 
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math>
−
| style="width:16%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math>
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| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math>
 
|-
 
|-
 
| style="border-top:1px solid black" |
 
| style="border-top:1px solid black" |
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<font face="courier new">
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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
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|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{Induced Transformation on Propositions}\!</math>
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|+ '''Table 65. Induced Transformation on Propositions'''
+
|- style="height:50px; background:ghostwhite"
 +
| style="width:20%" | <math>X^\bullet~\!</math>
 +
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math>
 +
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math>
 +
| style="width:20%; border-left:none"  | <math>\longleftarrow\!</math>
 +
| style="width:20%" | <math>U^\bullet~\!</math>
 
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
−
| ''X''<sup>&nbsp;&bull;</sup>
+
| rowspan="2" | <math>f_i (x, y)\!</math>
−
| colspan="3" |
+
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math>
−
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:80%"
+
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math>
−
| &larr;
+
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math>
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| ''F''&nbsp;=&nbsp;‹''f''&nbsp;,&nbsp;''g''›
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| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math>
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| &larr;
  −
|}
  −
| ''U''<sup>&nbsp;&bull;</sup>
   
|- style="background:ghostwhite"
 
|- style="background:ghostwhite"
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| rowspan="2" | ''f''<sub>''i''</sub>‹''x'',&nbsp;''y''›
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| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math>
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|
+
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math>
−
{| align="right" style="background:ghostwhite; text-align:right"
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| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math>
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| ''u'' =
   
|-
 
|-
−
| ''v'' =
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| valign="bottom" |
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|}
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<math>\begin{matrix}
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|
+
f_{0}
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{| align="center" style="background:ghostwhite; text-align:center"
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\\[2pt]
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| 1 1 0 0
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f_{1}
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f_{2}
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f_{3}
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f_{6}
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\\[2pt]
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f_{7}
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\end{matrix}</math>
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| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{(~)}
 +
\\[2pt]
 +
\texttt{(} x \texttt{)(} y \texttt{)}
 +
\\[2pt]
 +
\texttt{(} x \texttt{)~} y \texttt{~}
 +
\\[2pt]
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\texttt{(} x \texttt{)~~~}
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\\[2pt]
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\texttt{~} x \texttt{~(} y \texttt{)}
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\\[2pt]
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\texttt{~~~(} y \texttt{)}
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\\[2pt]
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\texttt{(} x \texttt{,~} y \texttt{)}
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\\[2pt]
 +
\texttt{(} x \texttt{~~} y \texttt{)}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
0~0~0~0
 +
\\[2pt]
 +
0~0~0~1
 +
\\[2pt]
 +
0~0~1~0
 +
\\[2pt]
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0~0~1~1
 +
\\[2pt]
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0~1~0~0
 +
\\[2pt]
 +
0~1~0~1
 +
\\[2pt]
 +
0~1~1~0
 +
\\[2pt]
 +
0~1~1~1
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{(~)}
 +
\\[2pt]
 +
\texttt{(~)}
 +
\\[2pt]
 +
\texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[2pt]
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\texttt{(} u \texttt{)(} v \texttt{)}
 +
\\[2pt]
 +
\texttt{(} u \texttt{,~} v \texttt{)}
 +
\\[2pt]
 +
\texttt{(} u \texttt{,~} v \texttt{)}
 +
\\[2pt]
 +
\texttt{(} u \texttt{~~} v \texttt{)}
 +
\\[2pt]
 +
\texttt{(} u \texttt{~~} v \texttt{)}
 +
\end{matrix}\!</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
f_{0}
 +
\\[2pt]
 +
f_{0}
 +
\\[2pt]
 +
f_{1}
 +
\\[2pt]
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f_{1}
 +
\\[2pt]
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f_{6}
 +
\\[2pt]
 +
f_{6}
 +
\\[2pt]
 +
f_{7}
 +
\\[2pt]
 +
f_{7}
 +
\end{matrix}</math>
 
|-
 
|-
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| 1 0 1 0
+
| valign="bottom" |
−
|}
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<math>\begin{matrix}
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|
+
f_{8}
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{| align="left" style="background:ghostwhite; text-align:left"
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\\[2pt]
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| = ''u''
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f_{9}
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|-
+
\\[2pt]
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| = ''v''
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f_{10}
−
|}
+
\\[2pt]
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| rowspan="2" | ''f''<sub>''j''</sub>‹''u'',&nbsp;''v''›
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f_{11}
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|- style="background:ghostwhite"
+
\\[2pt]
−
|
+
f_{12}
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{| align="right" style="background:ghostwhite; text-align:right"
+
\\[2pt]
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| ''x'' =
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f_{13}
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|-
+
\\[2pt]
−
| ''y'' =
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f_{14}
−
|}
+
\\[2pt]
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|
+
f_{15}
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{| align="center" style="background:ghostwhite; text-align:center"
+
\end{matrix}</math>
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| 1 1 1 0
+
| valign="bottom" |
−
|-
+
<math>\begin{matrix}
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| 1 0 0 1
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\texttt{~~} x \texttt{~~} y \texttt{~~}
−
|}
+
\\[2pt]
−
|
+
\texttt{((} x \texttt{,~} y \texttt{))}
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{| align="left" style="background:ghostwhite; text-align:left"
+
\\[2pt]
−
| = ''f''‹''u'',&nbsp;''v''›
+
\texttt{~~~~~} y \texttt{~~}
−
|-
+
\\[2pt]
−
| = ''g''‹''u'',&nbsp;''v''›
+
\texttt{~(} x \texttt{~(} y \texttt{))}
−
|}
+
\\[2pt]
−
|-
+
\texttt{~~} x \texttt{~~~~~}
−
|
+
\\[2pt]
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{| cellpadding="2"
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\texttt{((} x \texttt{)~} y \texttt{)~}
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| ''f''<sub>0</sub>
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\\[2pt]
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|-
+
\texttt{((} x \texttt{)(} y \texttt{))}
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| ''f''<sub>1</sub>
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\\[2pt]
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|-
+
\texttt{((~))}
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| ''f''<sub>2</sub>
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\end{matrix}</math>
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|-
+
| valign="bottom" |
−
| ''f''<sub>3</sub>
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<math>\begin{matrix}
−
|-
+
1~0~0~0
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| ''f''<sub>4</sub>
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\\[2pt]
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|-
+
1~0~0~1
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| ''f''<sub>5</sub>
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\\[2pt]
−
|-
+
1~0~1~0
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| ''f''<sub>6</sub>
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\\[2pt]
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|-
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1~0~1~1
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| ''f''<sub>7</sub>
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\\[2pt]
−
|}
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1~1~0~0
−
|
+
\\[2pt]
−
{| cellpadding="2"
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1~1~0~1
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| ()
+
\\[2pt]
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|-
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1~1~1~0
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| &nbsp;(''x'')(''y'')&nbsp;
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\\[2pt]
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|-
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1~1~1~1
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| &nbsp;(''x'')&nbsp;''y''&nbsp;&nbsp;
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\end{matrix}</math>
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|-
+
| valign="bottom" |
−
| &nbsp;(''x'')&nbsp;&nbsp;&nbsp;&nbsp;
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<math>\begin{matrix}
−
|-
+
\texttt{~~} u \texttt{~~} v \texttt{~~}
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| &nbsp;&nbsp;''x''&nbsp;(''y'')&nbsp;
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\\[2pt]
−
|-
+
\texttt{~~} u \texttt{~~} v \texttt{~~}
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| &nbsp;&nbsp;&nbsp;&nbsp;(''y'')&nbsp;
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\\[2pt]
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|-
+
\texttt{((} u \texttt{,~} v \texttt{))}
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| &nbsp;(''x'',&nbsp;''y'')&nbsp;
+
\\[2pt]
−
|-
+
\texttt{((} u \texttt{,~} v \texttt{))}
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| &nbsp;(''x''&nbsp;&nbsp;''y'')&nbsp;
+
\\[2pt]
−
|}
+
\texttt{((} u \texttt{)(} v \texttt{))}
−
|
+
\\[2pt]
−
{| cellpadding="2"
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\texttt{((} u \texttt{)(} v \texttt{))}
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| 0 0 0 0
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\\[2pt]
−
|-
+
\texttt{((~))}
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| 0 0 0 1
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\\[2pt]
−
|-
+
\texttt{((~))}
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| 0 0 1 0
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\end{matrix}</math>
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|-
+
| valign="bottom" |
−
| 0 0 1 1
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<math>\begin{matrix}
−
|-
+
f_{8}
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| 0 1 0 0
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\\[2pt]
−
|-
+
f_{8}
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| 0 1 0 1
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\\[2pt]
−
|-
+
f_{9}
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| 0 1 1 0
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\\[2pt]
−
|-
+
f_{9}
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| 0 1 1 1
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\\[2pt]
−
|}
+
f_{14}
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|
+
\\[2pt]
−
{| cellpadding="2"
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f_{14}
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| ()
+
\\[2pt]
−
|-
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f_{15}
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| ()
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\\[2pt]
−
|-
+
f_{15}
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| &nbsp;(''u'')(''v'')&nbsp;
+
\end{matrix}</math>
−
|-
  −
| &nbsp;(''u'')(''v'')&nbsp;
  −
|-
  −
| &nbsp;(''u'',&nbsp;''v'')&nbsp;
  −
|-
  −
| &nbsp;(''u'',&nbsp;''v'')&nbsp;
  −
|-
  −
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;
  −
|-
  −
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;
  −
|}
  −
|
  −
{| cellpadding="2"
  −
| ''f''<sub>0</sub>
  −
|-
  −
| ''f''<sub>0</sub>
  −
|-
  −
| ''f''<sub>1</sub>
  −
|-
  −
| ''f''<sub>1</sub>
  −
|-
  −
| ''f''<sub>6</sub>
  −
|-
  −
| ''f''<sub>6</sub>
  −
|-
  −
| ''f''<sub>7</sub>
  −
|-
  −
| ''f''<sub>7</sub>
  −
|}
  −
|-
  −
|
  −
{| cellpadding="2"
  −
| ''f''<sub>8</sub>
  −
|-
  −
| ''f''<sub>9</sub>
  −
|-
  −
| ''f''<sub>10</sub>
  −
|-
  −
| ''f''<sub>11</sub>
  −
|-
  −
| ''f''<sub>12</sub>
  −
|-
  −
| ''f''<sub>13</sub>
  −
|-
  −
| ''f''<sub>14</sub>
  −
|-
  −
| ''f''<sub>15</sub>
  −
|}
  −
|
  −
{| cellpadding="2"
  −
| &nbsp;&nbsp;''x''&nbsp;&nbsp;''y''&nbsp;&nbsp;
  −
|-
  −
| ((''x'',&nbsp;''y''))
  −
|-
  −
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''y''&nbsp;&nbsp;
  −
|-
  −
| &nbsp;(''x''&nbsp;(''y''))
  −
|-
  −
| &nbsp;&nbsp;''x''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
  −
|-
  −
| ((''x'')&nbsp;''y'')&nbsp;
  −
|-
  −
| ((''x'')(''y''))
  −
|-
  −
| (())
  −
|}
  −
|
  −
{| cellpadding="2"
  −
| 1 0 0 0
  −
|-
  −
| 1 0 0 1
  −
|-
  −
| 1 0 1 0
  −
|-
  −
| 1 0 1 1
  −
|-
  −
| 1 1 0 0
  −
|-
  −
| 1 1 0 1
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|-
  −
| 1 1 1 0
  −
|-
  −
| 1 1 1 1
  −
|}
  −
|
  −
{| cellpadding="2"
  −
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
  −
|-
  −
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
  −
|-
  −
| ((''u'',&nbsp;''v''))
  −
|-
  −
| ((''u'',&nbsp;''v''))
  −
|-
  −
| ((''u'')(''v''))
  −
|-
  −
| ((''u'')(''v''))
  −
|-
  −
| (())
  −
|-
  −
| (())
  −
|}
  −
|
  −
{| cellpadding="2"
  −
| ''f''<sub>8</sub>
  −
|-
  −
| ''f''<sub>8</sub>
  −
|-
  −
| ''f''<sub>9</sub>
  −
|-
  −
| ''f''<sub>9</sub>
  −
|-
  −
| ''f''<sub>14</sub>
  −
|-
  −
| ''f''<sub>14</sub>
  −
|-
  −
| ''f''<sub>15</sub>
  −
|-
  −
| ''f''<sub>15</sub>
  −
|}
   
|}
 
|}
−
</font>
      
<br>
 
<br>
Jon Awbrey
12,080

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