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==Truth Tables==
 
==Truth Tables==
 +
 +
===New Version===
  
 
<br>
 
<br>
  
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
−
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
+
|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
−
| width="15%" |
+
| width="25%" | <math>\mathcal{L}_1</math>
−
<p><math>\mathcal{L}_1</math></p>
+
| width="25%" | <math>\mathcal{L}_2</math>
−
<p><math>\text{Decimal}</math></p>
+
| width="25%" | <math>\mathcal{L}_3</math>
−
| width="15%" |
+
| width="25%" | <math>\mathcal{L}_4</math>
−
<p><math>\mathcal{L}_2</math></p>
 
−
<p><math>\text{Binary}</math></p>
 
−
| width="15%" |
 
−
<p><math>\mathcal{L}_3</math></p>
 
−
<p><math>\text{Vector}</math></p>
 
−
| width="15%" |
 
−
<p><math>\mathcal{L}_4</math></p>
 
−
<p><math>\text{Cactus}</math></p>
 
−
| width="25%" |
 
−
<p><math>\mathcal{L}_5</math></p>
 
−
<p><math>\text{English}</math></p>
 
−
| width="15%" |
 
−
<p><math>\mathcal{L}_6</math></p>
 
−
<p><math>\text{Ordinary}</math></p>
 
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
 
| &nbsp;
 
| &nbsp;
 
| align="right" | <math>p\colon\!</math>
 
| align="right" | <math>p\colon\!</math>
−
| <math>1~1~0~0</math>
+
| <math>1~1~1~1~0~0~0~0</math>
−
| &nbsp;
 
−
| &nbsp;
 
 
| &nbsp;
 
| &nbsp;
 
|- style="background:#f0f0ff"
 
|- style="background:#f0f0ff"
 
| &nbsp;
 
| &nbsp;
 
| align="right" | <math>q\colon\!</math>
 
| align="right" | <math>q\colon\!</math>
−
| <math>1~0~1~0</math>
+
| <math>1~1~0~0~1~1~0~0</math>
 
| &nbsp;
 
| &nbsp;
 +
|- style="background:#f0f0ff"
 
| &nbsp;
 
| &nbsp;
 +
| align="right" | <math>r\colon\!</math>
 +
| <math>1~0~1~0~1~0~1~0</math>
 
| &nbsp;
 
| &nbsp;
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_0
+
f_{104}
 
\\[4pt]
 
\\[4pt]
−
f_1
+
f_{148}
 
\\[4pt]
 
\\[4pt]
−
f_2
+
f_{146}
 
\\[4pt]
 
\\[4pt]
−
f_3
+
f_{97}
 
\\[4pt]
 
\\[4pt]
−
f_4
+
f_{134}
 
\\[4pt]
 
\\[4pt]
−
f_5
+
f_{73}
 
\\[4pt]
 
\\[4pt]
−
f_6
+
f_{41}
 
\\[4pt]
 
\\[4pt]
−
f_7
+
f_{22}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{0000}
+
f_{01101000}
 
\\[4pt]
 
\\[4pt]
−
f_{0001}
+
f_{10010100}
 
\\[4pt]
 
\\[4pt]
−
f_{0010}
+
f_{10010010}
 
\\[4pt]
 
\\[4pt]
−
f_{0011}
+
f_{01100001}
 
\\[4pt]
 
\\[4pt]
−
f_{0100}
+
f_{10000110}
 
\\[4pt]
 
\\[4pt]
−
f_{0101}
+
f_{01001001}
 
\\[4pt]
 
\\[4pt]
−
f_{0110}
+
f_{00101001}
 
\\[4pt]
 
\\[4pt]
−
f_{0111}
+
f_{00010110}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~0~0~0
+
0~1~1~0~1~0~0~0
 
\\[4pt]
 
\\[4pt]
−
0~0~0~1
+
1~0~0~1~0~1~0~0
 
\\[4pt]
 
\\[4pt]
−
0~0~1~0
+
1~0~0~1~0~0~1~0
 
\\[4pt]
 
\\[4pt]
−
0~0~1~1
+
0~1~1~0~0~0~0~1
 
\\[4pt]
 
\\[4pt]
−
0~1~0~0
+
1~0~0~0~0~1~1~0
 
\\[4pt]
 
\\[4pt]
−
0~1~0~1
+
0~1~0~0~1~0~0~1
 
\\[4pt]
 
\\[4pt]
−
0~1~1~0
+
0~0~1~0~1~0~0~1
 
\\[4pt]
 
\\[4pt]
−
0~1~1~1
+
0~0~0~1~0~1~1~0
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(~)
+
( p , q , r )
 
\\[4pt]
 
\\[4pt]
−
(p)(q)
+
( p , q , (r))
 
\\[4pt]
 
\\[4pt]
−
(p)~q~
+
( p , (q), r )
 
\\[4pt]
 
\\[4pt]
−
(p)~~~
+
( p , (q), (r))
 
\\[4pt]
 
\\[4pt]
−
~p~(q)
+
((p), q , r )
 
\\[4pt]
 
\\[4pt]
−
~~~(q)
+
((p), q , (r))
 
\\[4pt]
 
\\[4pt]
−
(p,~q)
+
((p), (q), r )
 
\\[4pt]
 
\\[4pt]
−
(p~~q)
+
((p), (q), (r))
−
\end{matrix}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
\end{matrix}</math>
 
|-
 
|-
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_8
+
f_{233}
 
\\[4pt]
 
\\[4pt]
−
f_9
+
f_{214}
 
\\[4pt]
 
\\[4pt]
−
f_{10}
+
f_{182}
 
\\[4pt]
 
\\[4pt]
−
f_{11}
+
f_{121}
 
\\[4pt]
 
\\[4pt]
−
f_{12}
+
f_{158}
 
\\[4pt]
 
\\[4pt]
−
f_{13}
+
f_{109}
 
\\[4pt]
 
\\[4pt]
−
f_{14}
+
f_{107}
 
\\[4pt]
 
\\[4pt]
−
f_{15}
+
f_{151}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_{1000}
+
f_{11101001}
 
\\[4pt]
 
\\[4pt]
−
f_{1001}
+
f_{11010110}
 
\\[4pt]
 
\\[4pt]
−
f_{1010}
+
f_{10110110}
 
\\[4pt]
 
\\[4pt]
−
f_{1011}
+
f_{01111001}
 
\\[4pt]
 
\\[4pt]
−
f_{1100}
+
f_{10011110}
 
\\[4pt]
 
\\[4pt]
−
f_{1101}
+
f_{01101101}
 
\\[4pt]
 
\\[4pt]
−
f_{1110}
+
f_{01101011}
 
\\[4pt]
 
\\[4pt]
−
f_{1111}
+
f_{10010111}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
1~0~0~0
+
1~1~1~0~1~0~0~1
 
\\[4pt]
 
\\[4pt]
−
1~0~0~1
+
1~1~0~1~0~1~1~0
 
\\[4pt]
 
\\[4pt]
−
1~0~1~0
+
1~0~1~1~0~1~1~0
 
\\[4pt]
 
\\[4pt]
−
1~0~1~1
+
0~1~1~1~1~0~0~1
 
\\[4pt]
 
\\[4pt]
−
1~1~0~0
+
1~0~0~1~1~1~1~0
 
\\[4pt]
 
\\[4pt]
−
1~1~0~1
+
0~1~1~0~1~1~0~1
 
\\[4pt]
 
\\[4pt]
−
1~1~1~0
+
0~1~1~0~1~0~1~1
 
\\[4pt]
 
\\[4pt]
−
1~1~1~1
+
1~0~0~1~0~1~1~1
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~~p~~q~~
+
(((p), (q), (r)))
 
\\[4pt]
 
\\[4pt]
−
((p,~q))
+
(((p), (q), r ))
 
\\[4pt]
 
\\[4pt]
−
~~~~~q~~
+
(((p), q , (r)))
 
\\[4pt]
 
\\[4pt]
−
~(p~(q))
+
(((p), q , r ))
 
\\[4pt]
 
\\[4pt]
−
~~p~~~~~
+
(( p , (q), (r)))
 
\\[4pt]
 
\\[4pt]
−
((p)~q)~
+
(( p , (q), r ))
 
\\[4pt]
 
\\[4pt]
−
((p)(q))
+
(( p , q , (r)))
 
\\[4pt]
 
\\[4pt]
−
((~))
+
(( p , q , r ))
−
\end{matrix}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
\end{matrix}</math>
 
|}
 
|}
 +
 +
<br>
 +
 +
===Old Version===
  
 
<br>
 
<br>
Line 367: Line 288:
 
| 1 0 0 1 0 1 1 1
 
| 1 0 0 1 0 1 1 1
 
| <math>(( p , q , r ))\!</math>
 
| <math>(( p , q , r ))\!</math>
−
|}
 
−
 
−
<br>
 
−
 
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 
−
|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
 
−
|- style="background:#f0f0ff"
 
−
| width="25%" | <math>\mathcal{L}_1</math>
 
−
| width="25%" | <math>\mathcal{L}_2</math>
 
−
| width="25%" | <math>\mathcal{L}_3</math>
 
−
| width="25%" | <math>\mathcal{L}_4</math>
 
−
|- style="background:#f0f0ff"
 
−
| &nbsp;
 
−
| align="right" | <math>p\colon\!</math>
 
−
| <math>1~1~1~1~0~0~0~0</math>
 
−
| &nbsp;
 
−
|- style="background:#f0f0ff"
 
−
| &nbsp;
 
−
| align="right" | <math>q\colon\!</math>
 
−
| <math>1~1~0~0~1~1~0~0</math>
 
−
| &nbsp;
 
−
|- style="background:#f0f0ff"
 
−
| &nbsp;
 
−
| align="right" | <math>r\colon\!</math>
 
−
| <math>1~0~1~0~1~0~1~0</math>
 
−
| &nbsp;
 
−
|-
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|-
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
−
|
 
−
<math>\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}</math>
 
 
|}
 
|}
  

Revision as of 03:52, 23 August 2009

Logical Graphs

Truth Tables

New Version


\(\text{Table 1.}~~\text{Logical Boundaries and Their Complements}\)
\(\mathcal{L}_1\) \(\mathcal{L}_2\) \(\mathcal{L}_3\) \(\mathcal{L}_4\)
  \(p\colon\!\) \(1~1~1~1~0~0~0~0\)  
  \(q\colon\!\) \(1~1~0~0~1~1~0~0\)  
  \(r\colon\!\) \(1~0~1~0~1~0~1~0\)  

\(\begin{matrix} f_{104} \\[4pt] f_{148} \\[4pt] f_{146} \\[4pt] f_{97} \\[4pt] f_{134} \\[4pt] f_{73} \\[4pt] f_{41} \\[4pt] f_{22} \end{matrix}\)

\(\begin{matrix} f_{01101000} \\[4pt] f_{10010100} \\[4pt] f_{10010010} \\[4pt] f_{01100001} \\[4pt] f_{10000110} \\[4pt] f_{01001001} \\[4pt] f_{00101001} \\[4pt] f_{00010110} \end{matrix}\)

\(\begin{matrix} 0~1~1~0~1~0~0~0 \\[4pt] 1~0~0~1~0~1~0~0 \\[4pt] 1~0~0~1~0~0~1~0 \\[4pt] 0~1~1~0~0~0~0~1 \\[4pt] 1~0~0~0~0~1~1~0 \\[4pt] 0~1~0~0~1~0~0~1 \\[4pt] 0~0~1~0~1~0~0~1 \\[4pt] 0~0~0~1~0~1~1~0 \end{matrix}\)

\(\begin{matrix} ( p , q , r ) \\[4pt] ( p , q , (r)) \\[4pt] ( p , (q), r ) \\[4pt] ( p , (q), (r)) \\[4pt] ((p), q , r ) \\[4pt] ((p), q , (r)) \\[4pt] ((p), (q), r ) \\[4pt] ((p), (q), (r)) \end{matrix}\)

\(\begin{matrix} f_{233} \\[4pt] f_{214} \\[4pt] f_{182} \\[4pt] f_{121} \\[4pt] f_{158} \\[4pt] f_{109} \\[4pt] f_{107} \\[4pt] f_{151} \end{matrix}\)

\(\begin{matrix} f_{11101001} \\[4pt] f_{11010110} \\[4pt] f_{10110110} \\[4pt] f_{01111001} \\[4pt] f_{10011110} \\[4pt] f_{01101101} \\[4pt] f_{01101011} \\[4pt] f_{10010111} \end{matrix}\)

\(\begin{matrix} 1~1~1~0~1~0~0~1 \\[4pt] 1~1~0~1~0~1~1~0 \\[4pt] 1~0~1~1~0~1~1~0 \\[4pt] 0~1~1~1~1~0~0~1 \\[4pt] 1~0~0~1~1~1~1~0 \\[4pt] 0~1~1~0~1~1~0~1 \\[4pt] 0~1~1~0~1~0~1~1 \\[4pt] 1~0~0~1~0~1~1~1 \end{matrix}\)

\(\begin{matrix} (((p), (q), (r))) \\[4pt] (((p), (q), r )) \\[4pt] (((p), q , (r))) \\[4pt] (((p), q , r )) \\[4pt] (( p , (q), (r))) \\[4pt] (( p , (q), r )) \\[4pt] (( p , q , (r))) \\[4pt] (( p , q , r )) \end{matrix}\)


Old Version


\(\text{Table 1.}~~\text{Logical Boundaries and Their Complements}\)
\(\mathcal{L}_1\) \(\mathcal{L}_2\) \(\mathcal{L}_3\) \(\mathcal{L}_4\)
  \(p =\!\) 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 \((( p , q , r ))\!\)


Venn Diagrams

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