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

Revision as of 20:20, 12 September 2013

4,610 bytes added ,  20:20, 12 September 2013
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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
 
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
−
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{Induced Transformation on Propositions}\!</math>
+
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math>
 
|- style="height:50px; background:ghostwhite"
 
|- style="height:50px; background:ghostwhite"
 
| style="width:20%" | <math>X^\bullet~\!</math>
 
| style="width:20%" | <math>X^\bullet~\!</math>
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| valign="bottom" |
 
| valign="bottom" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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| valign="bottom" |
 
| valign="bottom" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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1~0~0~0
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\\[2pt]
 
1~0~0~0
 
1~0~0~0
 
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f_{15}
 
f_{15}
 
\end{matrix}</math>
 
\end{matrix}</math>
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|}
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<br>
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{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
 +
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math>
 +
|- style="height:50px; background:ghostwhite"
 +
| style="width:20%" | <math>X^\bullet~\!</math>
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| style="width:20%; border-right:none" | <math>\longleftarrow\!</math>
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| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math>
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| style="width:20%; border-left:none"  | <math>\longleftarrow\!</math>
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| style="width:20%" | <math>U^\bullet~\!</math>
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|- style="background:ghostwhite"
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| rowspan="2" | <math>f_i (x, y)\!</math>
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| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math>
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| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math>
 +
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math>
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| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math>
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|- style="background:ghostwhite"
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| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math>
 +
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math>
 +
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math>
 +
|-
 +
| <math>f_{0}\!</math>
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| <math>\texttt{(~)}\!</math>
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| <math>0~0~0~0\!</math>
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| <math>\texttt{(~)}\!</math>
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| <math>f_{0}\!</math>
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|-
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<math>\begin{matrix}
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f_{2}
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f_{8}
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\end{matrix}</math>
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<math>\begin{matrix}
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\texttt{(} x \texttt{)(} y \texttt{)}
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\\[2pt]
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\texttt{~} x \texttt{~~} y \texttt{~}
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\end{matrix}</math>
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<math>\begin{matrix}
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0~0~0~0
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\\[2pt]
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1~0~0~0
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\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
\texttt{(~)}
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\texttt{(} u \texttt{)(} v \texttt{)}
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\texttt{(} u \texttt{,~} v \texttt{)}
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\\[2pt]
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\texttt{~} u \texttt{~~} v \texttt{~}
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\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
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f_{0}
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f_{1}
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f_{6}
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\\[2pt]
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f_{8}
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\end{matrix}</math>
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|-
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| valign="bottom" |
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<math>\begin{matrix}
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f_{3}
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\\[2pt]
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f_{12}
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\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
\texttt{(} x \texttt{)}
 +
\\[2pt]
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\texttt{~} x \texttt{~}
 +
\end{matrix}</math>
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<math>\begin{matrix}
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\\[2pt]
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1~1~1~0
 +
\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
\texttt{~(} u \texttt{)(} v \texttt{)~}
 +
\\[2pt]
 +
\texttt{((} u \texttt{)(} v \texttt{))}
 +
\end{matrix}</math>
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| valign="bottom" |
 +
<math>\begin{matrix}
 +
f_{1}
 +
\\[2pt]
 +
f_{14}
 +
\end{matrix}</math>
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|-
 +
| valign="bottom" |
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<math>\begin{matrix}
 +
f_{6}
 +
\\[2pt]
 +
f_{9}
 +
\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
\texttt{~(} x \texttt{,~} y \texttt{)~}
 +
\\[2pt]
 +
\texttt{((} x \texttt{,~} y \texttt{))}
 +
\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
0~1~1~1
 +
\\[2pt]
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1~0~0~0
 +
\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
\texttt{(} u \texttt{~~} v \texttt{)}
 +
\\[2pt]
 +
\texttt{~} u \texttt{~~} v \texttt{~}
 +
\end{matrix}</math>
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| valign="bottom" |
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<math>\begin{matrix}
 +
f_{7}
 +
\\[2pt]
 +
f_{8}
 +
\end{matrix}</math>
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|-
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| valign="bottom" |
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<math>\begin{matrix}
 +
f_{5}
 +
\\[2pt]
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f_{10}
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\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{(} y \texttt{)}
 +
\\[2pt]
 +
\texttt{~} y \texttt{~}
 +
\end{matrix}</math>
 +
| valign="bottom" |
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<math>\begin{matrix}
 +
0~1~1~0
 +
\\[2pt]
 +
1~0~0~1
 +
\end{matrix}~\!</math>
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| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{~(} u \texttt{,~} v \texttt{)~}
 +
\\[2pt]
 +
\texttt{((} u \texttt{,~} v \texttt{))}
 +
\end{matrix}</math>
 +
| valign="bottom" |
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<math>\begin{matrix}
 +
f_{6}
 +
\\[2pt]
 +
f_{9}
 +
\end{matrix}</math>
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|-
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| valign="bottom" |
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<math>\begin{matrix}
 +
f_{7}
 +
\\[2pt]
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f_{11}
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\\[2pt]
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f_{13}
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\\[2pt]
 +
f_{14}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{~(} x \texttt{~~} y \texttt{)~}
 +
\\[2pt]
 +
\texttt{~(} x \texttt{~(} y \texttt{))}
 +
\\[2pt]
 +
\texttt{((} x \texttt{)~} y \texttt{)~}
 +
\\[2pt]
 +
\texttt{((} x \texttt{)(} y \texttt{))}
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
0~1~1~1
 +
\\[2pt]
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1~0~0~1
 +
\\[2pt]
 +
1~1~1~0
 +
\\[2pt]
 +
1~1~1~1
 +
\end{matrix}</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
\texttt{~(} u \texttt{~~} v \texttt{)~}
 +
\\[2pt]
 +
\texttt{((} u \texttt{,~} v \texttt{))}
 +
\\[2pt]
 +
\texttt{((} u \texttt{)(} v \texttt{))}
 +
\\[2pt]
 +
\texttt{((~))}
 +
\end{matrix}\!</math>
 +
| valign="bottom" |
 +
<math>\begin{matrix}
 +
f_{7}
 +
\\[2pt]
 +
f_{9}
 +
\\[2pt]
 +
f_{14}
 +
\\[2pt]
 +
f_{15}
 +
\end{matrix}</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
| <math>1~1~1~1\!</math>
 +
| <math>\texttt{((~))}\!</math>
 +
| <math>f_{15}\!</math>
 
|}
 
|}
  
Jon Awbrey
12,122

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