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Revision as of 15:39, 9 September 2013
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<p><math>\mathcal{L}_1</math></p>+<p><math>\text{Decimal}</math></p>+<p><math>\mathcal{L}_2</math></p>+<p><math>\text{Binary}</math></p>+<p><math>\mathcal{L}_3</math></p>
−<p><math>\text{Vector}</math></p>
−<p><math>\mathcal{L}_4</math></p>
−<p><math>\text{Cactus}</math></p>
−<p><math>\mathcal{L}_5</math></p>
−<p><math>\text{English}</math></p>
−<p><math>\mathcal{L}_6</math></p>
−<p><math>\text{Ordinary}</math></p>
−f_0+f_1
−f_2
−f_3
−f_4
−f_5
−f_6
−f_7
−f_8+f_9
−
<br>
<br>
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
+{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"
−|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
+|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
−|- style="background:#f0f0ff"
+|- style="background:ghostwhite"
−| width="15%" |
+| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math>
−| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math>
−| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math>
−| width="15%" |
+| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math>
−| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math>
−| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math>
−| width="15%" |
+|- style="background:ghostwhite"
−| width="15%" |
−| width="25%" |
−| width="15%" |
−|- style="background:#f0f0ff"
|
|
| align="right" | <math>x\colon\!</math>
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
| <math>1~1~0~0\!</math>
−|
+| || ||
−|
+|- style="background:ghostwhite"
−|
−|- style="background:#f0f0ff"
|
|
| align="right" | <math>y\colon\!</math>
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
| <math>1~0~1~0\!</math>
−|
+| || ||
−|
−|
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}
−\\[4pt]
+\end{matrix}</math>
−\\[4pt]
−\\[4pt]
−\\[4pt]
−\\[4pt]
−\\[4pt]
−\\[4pt]
−\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−f_{0000}
+f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}
−\\[4pt]
+\end{matrix}</math>
−f_{0001}
−\\[4pt]
−f_{0010}
−\\[4pt]
−f_{0011}
−\\[4pt]
−f_{0100}
−\\[4pt]
−f_{0101}
−\\[4pt]
−f_{0110}
−\\[4pt]
−f_{0111}
−\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−0~0~0~0
+0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1
−\\[4pt]
−0~0~0~1
−\\[4pt]
−0~0~1~0
−\\[4pt]
−0~0~1~1
−\\[4pt]
−0~1~0~0
−\\[4pt]
−0~1~0~1
−\\[4pt]
−0~1~1~0
−\\[4pt]
−0~1~1~1
\end{matrix}\!</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−(~)
+\texttt{(~)}
−\\[4pt]
+\\
−(x)(y)
+\texttt{(} x \texttt{)(} y \texttt{)}
−\\[4pt]
+\\
−(x)~y~
+\texttt{(} x \texttt{)~} y \texttt{~}
−\\[4pt]
+\\
−(x)~~~
+\texttt{(} x \texttt{)~~~}
−\\[4pt]
+\\
−~x~(y)
+\texttt{~} x \texttt{~(} y \texttt{)}
−\\[4pt]
+\\
−~~~(y)
+\texttt{~~~(} y \texttt{)}
−\\[4pt]
+\\
−(x,~y)
+\texttt{(} x \texttt{,~} y \texttt{)}
−\\[4pt]
+\\
−(x~~y)
+\texttt{(} x \texttt{~~} y \texttt{)}
−\end{matrix}\!</math>
+\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\text{false}
\text{false}
−\\[4pt]
+\\
\text{neither}~ x ~\text{nor}~ y
\text{neither}~ x ~\text{nor}~ y
−\\[4pt]
+\\
y ~\text{without}~ x
y ~\text{without}~ x
−\\[4pt]
+\\
\text{not}~ x
\text{not}~ x
−\\[4pt]
+\\
x ~\text{without}~ y
x ~\text{without}~ y
−\\[4pt]
+\\
\text{not}~ y
\text{not}~ y
−\\[4pt]
+\\
x ~\text{not equal to}~ y
x ~\text{not equal to}~ y
−\\[4pt]
+\\
\text{not both}~ x ~\text{and}~ y
\text{not both}~ x ~\text{and}~ y
−\end{matrix}\!</math>
+\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
0
0
−\\[4pt]
+\\
\lnot x \land \lnot y
\lnot x \land \lnot y
−\\[4pt]
+\\
\lnot x \land y
\lnot x \land y
−\\[4pt]
+\\
\lnot x
\lnot x
−\\[4pt]
+\\
x \land \lnot y
x \land \lnot y
−\\[4pt]
+\\
\lnot y
\lnot y
−\\[4pt]
+\\
x \ne y
x \ne y
−\\[4pt]
+\\
\lnot x \lor \lnot y
\lnot x \lor \lnot y
−\end{matrix}\!</math>
+\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}
−\\[4pt]
+\end{matrix}</math>
−\\[4pt]
−f_{10}
−\\[4pt]
−f_{11}
−\\[4pt]
−f_{12}
−\\[4pt]
−f_{13}
−\\[4pt]
−f_{14}
−\\[4pt]
−f_{15}
−\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−f_{1000}
+f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}
−\\[4pt]
−f_{1001}
−\\[4pt]
−f_{1010}
−\\[4pt]
−f_{1011}
−\\[4pt]
−f_{1100}
−\\[4pt]
−f_{1101}
−\\[4pt]
−f_{1110}
−\\[4pt]
−f_{1111}
\end{matrix}\!</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−1~0~0~0
+1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1
−\\[4pt]
+\end{matrix}</math>
−1~0~0~1
−\\[4pt]
−1~0~1~0
−\\[4pt]
−1~0~1~1
−\\[4pt]
−1~1~0~0
−\\[4pt]
−1~1~0~1
−\\[4pt]
−1~1~1~0
−\\[4pt]
−1~1~1~1
−\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
−~~x~~y~~
+\texttt{~~} x \texttt{~~} y \texttt{~~}
−\\[4pt]
+\\
−((x,~y))
+\texttt{((} x \texttt{,~} y \texttt{))}
−\\[4pt]
+\\
−~~~~~y~~
+\texttt{~~~~~} y \texttt{~~}
−\\[4pt]
+\\
−~(x~(y))
+\texttt{~(} x \texttt{~(} y \texttt{))}
−\\[4pt]
+\\
−~~x~~~~~
+\texttt{~~} x \texttt{~~~~~}
−\\[4pt]
+\\
−((x)~y)~
+\texttt{((} x \texttt{)~} y \texttt{)~}
−\\[4pt]
+\\
−((x)(y))
+\texttt{((} x \texttt{)(} y \texttt{))}
−\\[4pt]
+\\
−((~))
+\texttt{((~))}
−\end{matrix}\!</math>
+\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x ~\text{and}~ y
x ~\text{and}~ y
−\\[4pt]
+\\
x ~\text{equal to}~ y
x ~\text{equal to}~ y
−\\[4pt]
+\\
y
y
−\\[4pt]
+\\
\text{not}~ x ~\text{without}~ y
\text{not}~ x ~\text{without}~ y
−\\[4pt]
+\\
x
x
−\\[4pt]
+\\
\text{not}~ y ~\text{without}~ x
\text{not}~ y ~\text{without}~ x
−\\[4pt]
+\\
x ~\text{or}~ y
x ~\text{or}~ y
−\\[4pt]
+\\
\text{true}
\text{true}
−\end{matrix}\!</math>
+\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x \land y
x \land y
−\\[4pt]
+\\
x = y
x = y
−\\[4pt]
+\\
y
y
−\\[4pt]
+\\
x \Rightarrow y
x \Rightarrow y
−\\[4pt]
+\\
x
x
−\\[4pt]
+\\
x \Leftarrow y
x \Leftarrow y
−\\[4pt]
+\\
x \lor y
x \lor y
−\\[4pt]
+\\
1
1
−\end{matrix}\!</math>
+\end{matrix}</math>
|}
|}
