Difference between revisions of "User:Jon Awbrey/MNO"

MyWikiBiz, Author Your Legacy — Wednesday December 04, 2024
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(tweak a bit to raise back to consciousness ...)
 
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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%" |
+
| <math>\mathcal{L}_1</math>
−
<p><math>\mathcal{L}_1</math></p>
+
| <math>\mathcal{L}_2</math>
−
<p><math>\text{Decimal}</math></p>
+
| <math>\mathcal{L}_3</math>
−
| width="15%" |
+
| <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]
 
−
f_1
 
−
\\[4pt]
 
−
f_2
 
−
\\[4pt]
 
−
f_3
 
−
\\[4pt]
 
−
f_4
 
−
\\[4pt]
 
−
f_5
 
−
\\[4pt]
 
−
f_6
 
−
\\[4pt]
 
−
f_7
 
−
\end{matrix}</math>
 
−
|
 
−
<math>\begin{matrix}
 
−
f_{0000}
 
 
\\[4pt]
 
\\[4pt]
−
f_{0001}
+
f_{148}
 
\\[4pt]
 
\\[4pt]
−
f_{0010}
+
f_{146}
 
\\[4pt]
 
\\[4pt]
−
f_{0011}
+
f_{97}
 
\\[4pt]
 
\\[4pt]
−
f_{0100}
+
f_{134}
 
\\[4pt]
 
\\[4pt]
−
f_{0101}
+
f_{73}
 
\\[4pt]
 
\\[4pt]
−
f_{0110}
+
f_{41}
 
\\[4pt]
 
\\[4pt]
−
f_{0111}
+
f_{22}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
0~0~0~0
+
f_{01101000}
 
\\[4pt]
 
\\[4pt]
−
0~0~0~1
+
f_{10010100}
 
\\[4pt]
 
\\[4pt]
−
0~0~1~0
+
f_{10010010}
 
\\[4pt]
 
\\[4pt]
−
0~0~1~1
+
f_{01100001}
 
\\[4pt]
 
\\[4pt]
−
0~1~0~0
+
f_{10000110}
 
\\[4pt]
 
\\[4pt]
−
0~1~0~1
+
f_{01001001}
 
\\[4pt]
 
\\[4pt]
−
0~1~1~0
+
f_{00101001}
 
\\[4pt]
 
\\[4pt]
−
0~1~1~1
+
f_{00010110}
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(~)
+
0~1~1~0~1~0~0~0
 
\\[4pt]
 
\\[4pt]
−
(p)(q)
+
1~0~0~1~0~1~0~0
 
\\[4pt]
 
\\[4pt]
−
(p)~q~
+
1~0~0~1~0~0~1~0
 
\\[4pt]
 
\\[4pt]
−
(p)~~~
+
0~1~1~0~0~0~0~1
 
\\[4pt]
 
\\[4pt]
−
~p~(q)
+
1~0~0~0~0~1~1~0
 
\\[4pt]
 
\\[4pt]
−
~~~(q)
+
0~1~0~0~1~0~0~1
 
\\[4pt]
 
\\[4pt]
−
(p,~q)
+
0~0~1~0~1~0~0~1
 
\\[4pt]
 
\\[4pt]
−
(p~~q)
+
0~0~0~1~0~1~1~0
 
\end{matrix}</math>
 
\end{matrix}</math>
 
|
 
|
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\text{false}
+
\texttt{(~p~,~q~,~r~)}
 
\\[4pt]
 
\\[4pt]
−
\text{neither}~ p ~\text{nor}~ q
+
\texttt{(~p~,~q~,(r))}
 
\\[4pt]
 
\\[4pt]
−
q ~\text{without}~ p
+
\texttt{(~p~,(q),~r~)}
 
\\[4pt]
 
\\[4pt]
−
\text{not}~ p
+
\texttt{(~p~,(q),(r))}
 
\\[4pt]
 
\\[4pt]
−
p ~\text{without}~ q
+
\texttt{((p),~q~,~r~)}
 
\\[4pt]
 
\\[4pt]
−
\text{not}~ q
+
\texttt{((p),~q~,(r))}
 
\\[4pt]
 
\\[4pt]
−
p ~\text{not equal to}~ q
+
\texttt{((p),(q),~r~)}
 
\\[4pt]
 
\\[4pt]
−
\text{not both}~ p ~\text{and}~ q
+
\texttt{((p),(q),(r))}
−
\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~~
+
\texttt{(((p),(q),(r)))}
 
\\[4pt]
 
\\[4pt]
−
((p,~q))
+
\texttt{(((p),(q),~r~))}
 
\\[4pt]
 
\\[4pt]
−
~~~~~q~~
+
\texttt{(((p),~q~,(r)))}
 
\\[4pt]
 
\\[4pt]
−
~(p~(q))
+
\texttt{(((p),~q~,~r~))}
 
\\[4pt]
 
\\[4pt]
−
~~p~~~~~
+
\texttt{((~p~,(q),(r)))}
 
\\[4pt]
 
\\[4pt]
−
((p)~q)~
+
\texttt{((~p~,(q),~r~))}
 
\\[4pt]
 
\\[4pt]
−
((p)(q))
+
\texttt{((~p~,~q~,(r)))}
 
\\[4pt]
 
\\[4pt]
−
((~))
+
\texttt{((~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>
 
|}
 
|}
Line 262: Line 179:
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f0f0ff; font-weight:bold; text-align:center; width:80%"
+
===Old Version===
−
|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
 
−
| width="20%" | <math>\mathcal{L}_1</math>
 
−
| width="20%" | <math>\mathcal{L}_2</math>
 
−
| width="20%" | <math>\mathcal{L}_3</math>
 
−
| width="20%" | <math>\mathcal{L}_4</math>
 
−
|-
 
−
| Decimal
 
−
| Binary
 
−
| Sequential
 
−
| Parenthetical
 
−
|-
 
−
| &nbsp;
 
−
| align="right" | <math>p =\!</math>
 
−
| 1 1 1 1 0 0 0 0
 
−
| &nbsp;
 
−
|-
 
−
| &nbsp;
 
−
| align="right" | <math>q =\!</math>
 
−
| 1 1 0 0 1 1 0 0
 
−
| &nbsp;
 
−
|-
 
−
| &nbsp;
 
−
| align="right" | <math>r =\!</math>
 
−
| 1 0 1 0 1 0 1 0
 
−
| &nbsp;
 
−
|}
 
−
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:80%"
 
−
|-
 
−
| width="20%" | <math>f_{104}\!</math>
 
−
| width="20%" | <math>f_{01101000}\!</math>
 
−
| width="20%" | 0 1 1 0 1 0 0 0
 
−
| width="20%" | <math>( p , q , r )\!</math>
 
−
|-
 
−
| <math>f_{148}\!</math>
 
−
| <math>f_{10010100}\!</math>
 
−
| 1 0 0 1 0 1 0 0
 
−
| <math>( p , q , (r))\!</math>
 
−
|-
 
−
| <math>f_{146}\!</math>
 
−
| <math>f_{10010010}\!</math>
 
−
| 1 0 0 1 0 0 1 0
 
−
| <math>( p , (q), r )\!</math>
 
−
|-
 
−
| <math>f_{97}\!</math>
 
−
| <math>f_{01100001}\!</math>
 
−
| 0 1 1 0 0 0 0 1
 
−
| <math>( p , (q), (r))\!</math>
 
−
|-
 
−
| <math>f_{134}\!</math>
 
−
| <math>f_{10000110}\!</math>
 
−
| 1 0 0 0 0 1 1 0
 
−
| <math>((p), q , r )\!</math>
 
−
|-
 
−
| <math>f_{73}\!</math>
 
−
| <math>f_{01001001}\!</math>
 
−
| 0 1 0 0 1 0 0 1
 
−
| <math>((p), q , (r))\!</math>
 
−
|-
 
−
| <math>f_{41}\!</math>
 
−
| <math>f_{00101001}\!</math>
 
−
| 0 0 1 0 1 0 0 1
 
−
| <math>((p), (q), r )\!</math>
 
−
|-
 
−
| <math>f_{22}\!</math>
 
−
| <math>f_{00010110}\!</math>
 
−
| 0 0 0 1 0 1 1 0
 
−
| <math>((p), (q), (r))\!</math>
 
−
|}
 
−
{|  align="center" border="1" cellpadding="4" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:80%"
 
−
|-
 
−
| width="20%" | <math>f_{233}\!</math>
 
−
| width="20%" | <math>f_{11101001}\!</math>
 
−
| width="20%" | 1 1 1 0 1 0 0 1
 
−
| width="20%" | <math>(((p), (q), (r)))\!</math>
 
−
|-
 
−
| <math>f_{214}\!</math>
 
−
| <math>f_{11010110}\!</math>
 
−
| 1 1 0 1 0 1 1 0
 
−
| <math>(((p), (q), r ))\!</math>
 
−
|-
 
−
| <math>f_{182}\!</math>
 
−
| <math>f_{10110110}\!</math>
 
−
| 1 0 1 1 0 1 1 0
 
−
| <math>(((p), q , (r)))\!</math>
 
−
|-
 
−
| <math>f_{121}\!</math>
 
−
| <math>f_{01111001}\!</math>
 
−
| 0 1 1 1 1 0 0 1
 
−
| <math>(((p), q , r ))\!</math>
 
−
|-
 
−
| <math>f_{158}\!</math>
 
−
| <math>f_{10011110}\!</math>
 
−
| 1 0 0 1 1 1 1 0
 
−
| <math>(( p , (q), (r)))\!</math>
 
−
|-
 
−
| <math>f_{109}\!</math>
 
−
| <math>f_{01101101}\!</math>
 
−
| 0 1 1 0 1 1 0 1
 
−
| <math>(( p , (q), r ))\!</math>
 
−
|-
 
−
| <math>f_{107}\!</math>
 
−
| <math>f_{01101011}\!</math>
 
−
| 0 1 1 0 1 0 1 1
 
−
| <math>(( p , q , (r)))\!</math>
 
−
|-
 
−
| <math>f_{151}\!</math>
 
−
| <math>f_{10010111}\!</math>
 
−
| 1 0 0 1 0 1 1 1
 
−
| <math>(( p , q , r ))\!</math>
 
−
|}
 
−
 
 
−
<br>
 
−
 
 
−
==Work Area==
 
  
 
<br>
 
<br>
Line 382: Line 185:
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f0f0ff; font-weight:bold; text-align:center; width:90%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f0f0ff; font-weight:bold; text-align:center; width:90%"
 
|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
 
|+ <math>\text{Table 1.}~~\text{Logical Boundaries and Their Complements}</math>
−
| width="20%" | <math>\mathcal{L}_1</math>
+
| width="25%" | <math>\mathcal{L}_1</math>
−
| width="20%" | <math>\mathcal{L}_2</math>
+
| width="25%" | <math>\mathcal{L}_2</math>
−
| width="20%" | <math>\mathcal{L}_3</math>
+
| width="25%" | <math>\mathcal{L}_3</math>
−
| width="20%" | <math>\mathcal{L}_4</math>
+
| width="25%" | <math>\mathcal{L}_4</math>
 
|-
 
|-
 
| &nbsp;
 
| &nbsp;
Line 404: Line 207:
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
|-
 
|-
−
| width="20%" | <math>f_{104}\!</math>
+
| width="25%" | <math>f_{104}\!</math>
−
| width="20%" | <math>f_{01101000}\!</math>
+
| width="25%" | <math>f_{01101000}\!</math>
−
| width="20%" | 0 1 1 0 1 0 0 0
+
| width="25%" | 0 1 1 0 1 0 0 0
−
| width="20%" | <math>( p , q , r )\!</math>
+
| width="25%" | <math>( p , q , r )\!</math>
 
|-
 
|-
 
| <math>f_{148}\!</math>
 
| <math>f_{148}\!</math>
Line 446: Line 249:
 
{|  align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
{|  align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
 
|-
 
|-
−
| width="20%" | <math>f_{233}\!</math>
+
| width="25%" | <math>f_{233}\!</math>
−
| width="20%" | <math>f_{11101001}\!</math>
+
| width="25%" | <math>f_{11101001}\!</math>
−
| width="20%" | 1 1 1 0 1 0 0 1
+
| width="25%" | 1 1 1 0 1 0 0 1
−
| width="20%" | <math>(((p), (q), (r)))\!</math>
+
| width="25%" | <math>(((p), (q), (r)))\!</math>
 
|-
 
|-
 
| <math>f_{214}\!</math>
 
| <math>f_{214}\!</math>
Line 489: Line 292:
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
+
==Venn Diagrams==
−
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
+
 
−
|- style="background:#f0f0ff"
+
===New Version===
−
| width="15%" | <math>\mathcal{L}_1</math>
+
 
−
| width="15%" | <math>\mathcal{L}_2</math>
+
{| align="center" cellpadding="10" style="text-align:center"
−
| width="15%" | <math>\mathcal{L}_3</math>
 
−
| width="15%" | <math>\mathcal{L}_4</math>
 
−
|- style="background:#f0f0ff"
 
−
| &nbsp;
 
−
| align="right" | <math>p\colon\!</math>
 
−
| <math>1~1~0~0\!</math>
 
−
| &nbsp;
 
−
|- style="background:#f0f0ff"
 
−
| &nbsp;
 
−
| align="right" | <math>q\colon\!</math>
 
−
| <math>1~0~1~0\!</math>
 
−
| &nbsp;
 
−
|-
 
 
|
 
|
−
<math>\begin{matrix}
+
<p>[[Image:Venn Diagram (P,Q,R).jpg|500px]]</p>
−
f_0
+
<p><math>\text{Figure 2.}~~\texttt{(p, q, r)}</math>
−
\\[4pt]
+
|}
−
f_1
+
 
−
\\[4pt]
+
{| align="center" cellpadding="10" style="text-align:center"
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f_2
 
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f_3
 
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f_4
 
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f_6
 
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f_7
 
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\end{matrix}</math>
 
 
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<math>\begin{matrix}
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<p>[[Image:Venn Diagram ((P),(Q),(R)).jpg|500px]]</p>
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f_{0000}
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<p><math>\text{Figure 3.}~~\texttt{((p),(q),(r))}</math>
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|}
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f_{0001}
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===Old Version===
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f_{0010}
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\end{matrix}</math>
 
 
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<math>\begin{matrix}
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<p>[[Image:Minimal Negation Operator 1.jpg|500px]]</p>
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0~0~0~0
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<p><math>\text{Figure 2.} ~~ \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}</math>
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<math>\begin{matrix}
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<p>[[Image:Minimal Negation Operator 2.jpg|500px]]</p>
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~~p~~q~~
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<p><math>\text{Figure 3.} ~~ \texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}</math>
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<br>
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*
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==Venn Diagrams==
 

Latest revision as of 16:10, 28 August 2017

Logical Graphs

Truth Tables

New Version


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

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

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

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

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

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

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

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

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


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

New Version

Venn Diagram (P,Q,R).jpg

\(\text{Figure 2.}~~\texttt{(p, q, r)}\)

Venn Diagram ((P),(Q),(R)).jpg

\(\text{Figure 3.}~~\texttt{((p),(q),(r))}\)

Old Version

Minimal Negation Operator 1.jpg

\(\text{Figure 2.} ~~ \texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}\)

Minimal Negation Operator 2.jpg

\(\text{Figure 3.} ~~ \texttt{((} p \texttt{),(} q \texttt{),(} r \texttt{))}\)

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