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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
, 20:20, 12 September 2013update
{| 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>
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
0~0~0~0
\\[2pt]
0~0~0~0
0~0~0~0
\\[2pt]
\\[2pt]
0~0~0~1
0~0~0~1
\\[2pt]
\\[2pt]
0~0~1~0
0~0~0~1
\\[2pt]
\\[2pt]
0~0~1~1
0~1~1~0
\\[2pt]
\\[2pt]
0~1~0~0
0~1~1~0
\\[2pt]
\\[2pt]
0~1~0~1
0~1~1~1
\\[2pt]
\\[2pt]
0~1~1~1
0~1~1~1
| valign="bottom" |
| valign="bottom" |
<math>\begin{matrix}
<math>\begin{matrix}
1~0~0~0
\\[2pt]
1~0~0~0
1~0~0~0
\\[2pt]
\\[2pt]
1~0~0~1
1~0~0~1
\\[2pt]
\\[2pt]
1~0~1~0
1~0~0~1
\\[2pt]
\\[2pt]
1~0~1~1
1~1~1~0
\\[2pt]
\\[2pt]
1~1~0~0
1~1~1~0
\\[2pt]
\\[2pt]
1~1~0~1
1~1~1~1
1~1~1~0
\\[2pt]
\\[2pt]
1~1~1~1
1~1~1~1
f_{15}
f_{15}
\end{matrix}</math>
\end{matrix}</math>
|}
<br>
{| 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>
| 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"
| rowspan="2" | <math>f_i (x, y)\!</math>
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math>
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math>
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math>
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math>
|- style="background:ghostwhite"
| 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>
| <math>\texttt{(~)}\!</math>
| <math>0~0~0~0\!</math>
| <math>\texttt{(~)}\!</math>
| <math>f_{0}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{1}
\\[2pt]
f_{2}
\\[2pt]
f_{4}
\\[2pt]
f_{8}
\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}
0~0~0~0
\\[2pt]
0~0~0~1
\\[2pt]
0~1~1~0
\\[2pt]
1~0~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\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_{1}
\\[2pt]
f_{6}
\\[2pt]
f_{8}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{3}
\\[2pt]
f_{12}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} x \texttt{)}
\\[2pt]
\texttt{~} x \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~1
\\[2pt]
1~1~1~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} u \texttt{)(} v \texttt{)~}
\\[2pt]
\texttt{((} u \texttt{)(} v \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1}
\\[2pt]
f_{14}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{6}
\\[2pt]
f_{9}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} x \texttt{,~} y \texttt{)~}
\\[2pt]
\texttt{((} x \texttt{,~} y \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~1
\\[2pt]
1~0~0~0
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} u \texttt{~~} v \texttt{)}
\\[2pt]
\texttt{~} u \texttt{~~} v \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[2pt]
f_{8}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{5}
\\[2pt]
f_{10}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(} y \texttt{)}
\\[2pt]
\texttt{~} y \texttt{~}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~0
\\[2pt]
1~0~0~1
\end{matrix}~\!</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{~(} u \texttt{,~} v \texttt{)~}
\\[2pt]
\texttt{((} u \texttt{,~} v \texttt{))}
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
f_{6}
\\[2pt]
f_{9}
\end{matrix}</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{7}
\\[2pt]
f_{11}
\\[2pt]
f_{13}
\\[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]
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>
|}
|}