User:Jon Awbrey/MNO

Truth Tables

\(\text{Table A1.}~~\text{Propositional Forms on Two Variables}\)

\(\mathcal{L}_1\) \(\text{Decimal}\)	\(\mathcal{L}_2\) \(\text{Binary}\)	\(\mathcal{L}_3\) \(\text{Vector}\)	\(\mathcal{L}_4\) \(\text{Cactus}\)	\(\mathcal{L}_5\) \(\text{English}\)	\(\mathcal{L}_6\) \(\text{Ordinary}\)
	\(p\colon\!\)	\(1~1~0~0\)
	\(q\colon\!\)	\(1~0~1~0\)
\(\begin{matrix} f_0 \\[4pt] f_1 \\[4pt] f_2 \\[4pt] f_3 \\[4pt] f_4 \\[4pt] f_5 \\[4pt] f_6 \\[4pt] f_7 \end{matrix}\)	\(\begin{matrix} f_{0000} \\[4pt] f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0011} \\[4pt] f_{0100} \\[4pt] f_{0101} \\[4pt] f_{0110} \\[4pt] f_{0111} \end{matrix}\)	\(\begin{matrix} 0~0~0~0 \\[4pt] 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~0~1~1 \\[4pt] 0~1~0~0 \\[4pt] 0~1~0~1 \\[4pt] 0~1~1~0 \\[4pt] 0~1~1~1 \end{matrix}\)	\(\begin{matrix} (~) \\[4pt] (p)(q) \\[4pt] (p)~q~ \\[4pt] (p)~~~ \\[4pt] ~p~(q) \\[4pt] ~~~(q) \\[4pt] (p,~q) \\[4pt] (p~~q) \end{matrix}\)	\(\begin{matrix} \text{false} \\[4pt] \text{neither}~ p ~\text{nor}~ q \\[4pt] q ~\text{without}~ p \\[4pt] \text{not}~ p \\[4pt] p ~\text{without}~ q \\[4pt] \text{not}~ q \\[4pt] p ~\text{not equal to}~ q \\[4pt] \text{not both}~ p ~\text{and}~ q \end{matrix}\)	\(\begin{matrix} 0 \\[4pt] \lnot p \land \lnot q \\[4pt] \lnot p \land q \\[4pt] \lnot p \\[4pt] p \land \lnot q \\[4pt] \lnot q \\[4pt] p \ne q \\[4pt] \lnot p \lor \lnot q \end{matrix}\)
\(\begin{matrix} f_8 \\[4pt] f_9 \\[4pt] f_{10} \\[4pt] f_{11} \\[4pt] f_{12} \\[4pt] f_{13} \\[4pt] f_{14} \\[4pt] f_{15} \end{matrix}\)	\(\begin{matrix} f_{1000} \\[4pt] f_{1001} \\[4pt] f_{1010} \\[4pt] f_{1011} \\[4pt] f_{1100} \\[4pt] f_{1101} \\[4pt] f_{1110} \\[4pt] f_{1111} \end{matrix}\)	\(\begin{matrix} 1~0~0~0 \\[4pt] 1~0~0~1 \\[4pt] 1~0~1~0 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~0 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \\[4pt] 1~1~1~1 \end{matrix}\)	\(\begin{matrix} ~~p~~q~~ \\[4pt] ((p,~q)) \\[4pt] ~~~~~q~~ \\[4pt] ~(p~(q)) \\[4pt] ~~p~~~~~ \\[4pt] ((p)~q)~ \\[4pt] ((p)(q)) \\[4pt] ((~)) \end{matrix}\)	\(\begin{matrix} p ~\text{and}~ q \\[4pt] p ~\text{equal to}~ q \\[4pt] q \\[4pt] \text{not}~ p ~\text{without}~ q \\[4pt] p \\[4pt] \text{not}~ q ~\text{without}~ p \\[4pt] p ~\text{or}~ q \\[4pt] \text{true} \end{matrix}\)	\(\begin{matrix} p \land q \\[4pt] p = q \\[4pt] q \\[4pt] p \Rightarrow q \\[4pt] p \\[4pt] p \Leftarrow q \\[4pt] p \lor q \\[4pt] 1 \end{matrix}\)

\(\text{Table 1.}~~\text{Logical Boundaries and Their Complements}\)

\(\mathcal{L}_1\)	\(\mathcal{L}_2\)	\(\mathcal{L}_3\)	\(\mathcal{L}_4\)
	\(p =\!\)	1 1 1 1 0 0 0 0
	\(q =\!\)	1 1 0 0 1 1 0 0
	\(r =\!\)	1 0 1 0 1 0 1 0

\(f_{104}\!\)	\(f_{01101000}\!\)	0 1 1 0 1 0 0 0	\(( p , q , r )\!\)
\(f_{148}\!\)	\(f_{10010100}\!\)	1 0 0 1 0 1 0 0	\(( p , q , (r))\!\)
\(f_{146}\!\)	\(f_{10010010}\!\)	1 0 0 1 0 0 1 0	\(( p , (q), r )\!\)
\(f_{97}\!\)	\(f_{01100001}\!\)	0 1 1 0 0 0 0 1	\(( p , (q), (r))\!\)
\(f_{134}\!\)	\(f_{10000110}\!\)	1 0 0 0 0 1 1 0	\(((p), q , r )\!\)
\(f_{73}\!\)	\(f_{01001001}\!\)	0 1 0 0 1 0 0 1	\(((p), q , (r))\!\)
\(f_{41}\!\)	\(f_{00101001}\!\)	0 0 1 0 1 0 0 1	\(((p), (q), r )\!\)
\(f_{22}\!\)	\(f_{00010110}\!\)	0 0 0 1 0 1 1 0	\(((p), (q), (r))\!\)

\(f_{233}\!\)	\(f_{11101001}\!\)	1 1 1 0 1 0 0 1	\((((p), (q), (r)))\!\)
\(f_{214}\!\)	\(f_{11010110}\!\)	1 1 0 1 0 1 1 0	\((((p), (q), r ))\!\)
\(f_{182}\!\)	\(f_{10110110}\!\)	1 0 1 1 0 1 1 0	\((((p), q , (r)))\!\)
\(f_{121}\!\)	\(f_{01111001}\!\)	0 1 1 1 1 0 0 1	\((((p), q , r ))\!\)
\(f_{158}\!\)	\(f_{10011110}\!\)	1 0 0 1 1 1 1 0	\((( p , (q), (r)))\!\)
\(f_{109}\!\)	\(f_{01101101}\!\)	0 1 1 0 1 1 0 1	\((( p , (q), r ))\!\)
\(f_{107}\!\)	\(f_{01101011}\!\)	0 1 1 0 1 0 1 1	\((( p , q , (r)))\!\)
\(f_{151}\!\)	\(f_{10010111}\!\)	1 0 0 1 0 1 1 1	\((( p , q , r ))\!\)

\(\text{Table 1.}~~\text{Logical Boundaries and Their Complements}\)

\(\mathcal{L}_1\)	\(\mathcal{L}_2\)	\(\mathcal{L}_3\)	\(\mathcal{L}_4\)
	\(p\colon\!\)	\(1~1~1~1~0~0~0~0\)
	\(q\colon\!\)	\(1~1~0~0~1~1~0~0\)
	\(r\colon\!\)	\(1~0~1~0~1~0~1~0\)
\(\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}\)	\(\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}\)	\(\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}\)	\(\begin{matrix} ( p , q , r ) \\[4pt] ( p , q , (r)) \\[4pt] ( p , (q), r ) \\[4pt] ( p , (q), (r)) \\[4pt] ((p), q , r ) \\[4pt] ((p), q , (r)) \\[4pt] ((p), (q), r ) \\[4pt] ((p), (q), (r)) \end{matrix}\)
\(\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}\)	\(\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}\)	\(\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}\)	\(\begin{matrix} (((p), (q), (r))) \\[4pt] (((p), (q), r )) \\[4pt] (((p), q , (r))) \\[4pt] (((p), q , r )) \\[4pt] (( p , (q), (r))) \\[4pt] (( p , (q), r )) \\[4pt] (( p , q , (r))) \\[4pt] (( p , q , r )) \end{matrix}\)