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Revision as of 12:30, 9 June 2008
, 12:30, 9 June 2008→Appendices @ PlanetMath : TeX Format: update
\multicolumn{7}{c}{Table A1. Propositional Forms on Two Variables} \\
\multicolumn{7}{c}{Table A1. Propositional Forms on Two Variables} \\
\hline
\hline
$\mathcal{L}_1$ & $\mathcal{L}_2$ &&
$\mathcal{L}_1$ &
$\mathcal{L}_3$ & $\mathcal{L}_4$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_5$ & $\mathcal{L}_6$ \\
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $x =$ & 1 1 0 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
\hline
\hline
$f_{0}$ & $f_{0000}$ & & 0 0 0 0 & $(~)$ & false & $0$ \\
$f_{0}$ &
$f_{1}$ & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$ & neither $x$ nor $y$ & $\lnot x \land \lnot y $ \\
$f_{0000}$ &&
$f_{2}$ & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$ & $y$ without $x$ & $\lnot x \land y$ \\
0 0 0 0 &
$f_{3}$ & $f_{0011}$ & & 0 0 1 1 & $(x)$ & not $x$ & $\lnot x$ \\
$(~)$ &
$f_{4}$ & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$ & $x$ without $y$ & $x \land \lnot y$ \\
$\operatorname{false}$ &
$f_{5}$ & $f_{0101}$ & & 0 1 0 1 & $(y)$ & not $y$ & $\lnot y$ \\
$0$ \\
$f_{6}$ & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$ & $x$ not equal to $y$ & $x \ne y$ \\
$f_{1}$ &
$f_{7}$ & $f_{0111}$ & & 0 1 1 1 & $(x\ y)$ & not both $x$ and $y$ & $\lnot x \lor \lnot y$ \\
$f_{0001}$ &&
0 0 0 1 &
$(x)(y)$ &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$f_{2}$ &
$f_{0010}$ &&
0 0 1 0 &
$(x)\ y$ &
$y\ \operatorname{without}\ x$ &
$\lnot x \land y$ \\
$f_{3}$ &
$f_{0011}$ &&
0 0 1 1 &
$(x)$ &
$\operatorname{not}\ x$ &
$\lnot x$ \\
$f_{4}$ &
$f_{0100}$ &&
0 1 0 0 &
$x\ (y)$ &
$x\ \operatorname{without}\ y$ &
$x \land \lnot y$ \\
$f_{5}$ &
$f_{0101}$ &&
0 1 0 1 &
$(y)$ &
$\operatorname{not}\ y$ &
$\lnot y$ \\
$f_{6}$ &
$f_{0110}$ &&
0 1 1 0 &
$(x,\ y)$ &
$x\ \operatorname{not~equal~to}\ y$ &
$x \ne y$ \\
$f_{7}$ &
$f_{0111}$ &&
0 1 1 1 &
$(x\ y)$ &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\lnot x \lor \lnot y$ \\
\hline
\hline
$f_{8}$ & $f_{1000}$ & & 1 0 0 0 & $x\ y$ & $x$ and $y$ & $x \land y$ \\
$f_{8}$ &
$f_{9}$ & $f_{1001}$ & & 1 0 0 1 & $((x,\ y))$ & $x$ equal to $y$ & $x = y$ \\
$f_{1000}$ &&
$f_{10}$ & $f_{1010}$ & & 1 0 1 0 & $y$ & $y$ & $y$ \\
1 0 0 0 &
$f_{11}$ & $f_{1011}$ & & 1 0 1 1 & $(x\ (y))$ & not $x$ without $y$ & $x \Rightarrow y$ \\
$x\ y$ &
$f_{12}$ & $f_{1100}$ & & 1 1 0 0 & $x$ & $x$ & $x$ \\
$x\ \operatorname{and}\ y$ &
$f_{13}$ & $f_{1101}$ & & 1 1 0 1 & $((x)\ y)$ & not $y$ without $x$ & $x \Leftarrow y$ \\
$x \land y$ \\
$f_{14}$ & $f_{1110}$ & & 1 1 1 0 & $((x)(y))$ & $x$ or $y$ & $x \lor y$ \\
$f_{9}$ &
$f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\
$f_{1001}$ &&
1 0 0 1 &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x = y$ \\
$f_{10}$ &
$f_{1010}$ &&
1 0 1 0 &
$y$ &
$y$ &
$y$ \\
$f_{11}$ &
$f_{1011}$ &&
1 0 1 1 &
$(x\ (y))$ &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$x \Rightarrow y$ \\
$f_{12}$ &
$f_{1100}$ &&
1 1 0 0 &
$x$ &
$x$ &
$x$ \\
$f_{13}$ &
$f_{1101}$ &&
1 1 0 1 &
$((x)\ y)$ &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$x \Leftarrow y$ \\
$f_{14}$ &
$f_{1110}$ &&
1 1 1 0 &
$((x)(y))$ &
$x\ \operatorname{or}\ y$ &
$x \lor y$ \\
$f_{15}$ &
$f_{1111}$ &&
1 1 1 1 &
$((~))$ &
$\operatorname{true}$ &
$1$ \\
\hline
\hline
\end{tabular}\end{quote}
\end{tabular}\end{quote}
\multicolumn{7}{c}{Table A2. Propositional Forms on Two Variables} \\
\multicolumn{7}{c}{Table A2. Propositional Forms on Two Variables} \\
\hline
\hline
$\mathcal{L}_1$ & $\mathcal{L}_2$ &&
$\mathcal{L}_1$ &
$\mathcal{L}_3$ & $\mathcal{L}_4$ &
$\mathcal{L}_2$ &&
$\mathcal{L}_5$ & $\mathcal{L}_6$ \\
$\mathcal{L}_3$ &
$\mathcal{L}_4$ &
$\mathcal{L}_5$ &
$\mathcal{L}_6$ \\
\hline
\hline
& & $x =$ & 1 1 0 0 & & & \\
& & $x =$ & 1 1 0 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
& & $y =$ & 1 0 1 0 & & & \\
\hline
\hline
$f_{0}$ & $f_{0000}$ & & 0 0 0 0 & $(~)$ & false & $0$ \\
$f_{0}$ &
$f_{0000}$ &&
0 0 0 0 &
$(~)$ &
$\operatorname{false}$ &
$0$ \\
\hline
\hline
$f_{1}$ & $f_{0001}$ & & 0 0 0 1 & $(x)(y)$ & neither $x$ nor $y$ & $\lnot x \land \lnot y $ \\
$f_{1}$ &
$f_{2}$ & $f_{0010}$ & & 0 0 1 0 & $(x)\ y$ & $y$ without $x$ & $\lnot x \land y$ \\
$f_{0001}$ &&
$f_{4}$ & $f_{0100}$ & & 0 1 0 0 & $x\ (y)$ & $x$ without $y$ & $x \land \lnot y$ \\
0 0 0 1 &
$f_{8}$ & $f_{1000}$ & & 1 0 0 0 & $x\ y$ & $x$ and $y$ & $x \land y$ \\
$(x)(y)$ &
$\operatorname{neither}\ x\ \operatorname{nor}\ y$ &
$\lnot x \land \lnot y$ \\
$f_{2}$ &
$f_{0010}$ &&
0 0 1 0 &
$(x)\ y$ &
$y\ \operatorname{without}\ x$ &
$\lnot x \land y$ \\
$f_{4}$ &
$f_{0100}$ &&
0 1 0 0 &
$x\ (y)$ &
$x\ \operatorname{without}\ y$ &
$x \land \lnot y$ \\
$f_{8}$ &
$f_{1000}$ &&
1 0 0 0 &
$x\ y$ &
$x\ \operatorname{and}\ y$ &
$x \land y$ \\
\hline
\hline
$f_{3}$ & $f_{0011}$ & & 0 0 1 1 & $(x)$ & not $x$ & $\lnot x$ \\
$f_{3}$ &
$f_{12}$ & $f_{1100}$ & & 1 1 0 0 & $x$ & $x$ & $x$ \\
$f_{0011}$ &&
0 0 1 1 &
$(x)$ &
$\operatorname{not}\ x$ &
$\lnot x$ \\
$f_{12}$ &
$f_{1100}$ &&
1 1 0 0 &
$x$ &
$x$ &
$x$ \\
\hline
\hline
$f_{6}$ & $f_{0110}$ & & 0 1 1 0 & $(x,\ y)$ & $x$ not equal to $y$ & $x \ne y$ \\
$f_{6}$ &
$f_{9}$ & $f_{1001}$ & & 1 0 0 1 & $((x,\ y))$ & $x$ equal to $y$ & $x = y$ \\
$f_{0110}$ &&
0 1 1 0 &
$(x,\ y)$ &
$x\ \operatorname{not~equal~to}\ y$ &
$x \ne y$ \\
$f_{9}$ &
$f_{1001}$ &&
1 0 0 1 &
$((x,\ y))$ &
$x\ \operatorname{equal~to}\ y$ &
$x = y$ \\
\hline
\hline
$f_{5}$ & $f_{0101}$ & & 0 1 0 1 & $(y)$ & not $y$ & $\lnot y$ \\
$f_{5}$ &
$f_{10}$ & $f_{1010}$ & & 1 0 1 0 & $y$ & $y$ & $y$ \\
$f_{0101}$ &&
0 1 0 1 &
$(y)$ &
$\operatorname{not}\ y$ &
$\lnot y$ \\
$f_{10}$ &
$f_{1010}$ &&
1 0 1 0 &
$y$ &
$y$ &
$y$ \\
\hline
\hline
$f_{7}$ & $f_{0111}$ & & 0 1 1 1 & $(x\ y)$ & not both $x$ and $y$ & $\lnot x \lor \lnot y$ \\
$f_{7}$ &
$f_{11}$ & $f_{1011}$ & & 1 0 1 1 & $(x\ (y))$ & not $x$ without $y$ & $x \Rightarrow y$ \\
$f_{0111}$ &&
$f_{13}$ & $f_{1101}$ & & 1 1 0 1 & $((x)\ y)$ & not $y$ without $x$ & $x \Leftarrow y$ \\
0 1 1 1 &
$f_{14}$ & $f_{1110}$ & & 1 1 1 0 & $((x)(y))$ & $x$ or $y$ & $x \lor y$ \\
$(x\ y)$ &
$\operatorname{not~both}\ x\ \operatorname{and}\ y$ &
$\lnot x \lor \lnot y$ \\
$f_{11}$ &
$f_{1011}$ &&
1 0 1 1 &
$(x\ (y))$ &
$\operatorname{not}\ x\ \operatorname{without}\ y$ &
$x \Rightarrow y$ \\
$f_{13}$ &
$f_{1101}$ &&
1 1 0 1 &
$((x)\ y)$ &
$\operatorname{not}\ y\ \operatorname{without}\ x$ &
$x \Leftarrow y$ \\
$f_{14}$ &
$f_{1110}$ &&
1 1 1 0 &
$((x)(y))$ &
$x\ \operatorname{or}\ y$ &
$x \lor y$ \\
\hline
\hline
$f_{15}$ & $f_{1111}$ & & 1 1 1 1 & $((~))$ & true & $1$ \\
$f_{15}$ &
$f_{1111}$ &&
1 1 1 1 &
$((~))$ &
$\operatorname{true}$ &
$1$ \\
\hline
\hline
\end{tabular}\end{quote}
\end{tabular}\end{quote}
