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Revision as of 12:30, 9 June 2008
, 12:30, 9 June 2008→Appendices @ PlanetMath : TeX Format: update
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\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}
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\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}
