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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

Revision as of 06:22, 31 May 2008

2,109 bytes added ,  06:22, 31 May 2008
→‎Propositional Forms on Two Variables: sub new table
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| <p><math>(~)\!</math></p>
 
| <p><math>(~)\!</math></p>
 
| <p><math>\operatorname{false}</math></p>
 
| <p><math>\operatorname{false}</math></p>
−
| <p><math>1\!</math></p>
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| <p><math>0\!</math></p>
 
|-
 
|-
 
|
 
|
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
 
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
 
|+ '''Table 3.  <math>\operatorname{E}f</math> Expanded Over Ordinary Features <math>\{ x, y \}\!</math>'''
−
|- style="background:ghostwhite"
+
|- style="background:ghostwhite; height:36px"
−
| style="width:16%" | &nbsp;
+
| &nbsp;
−
| style="width:16%" | <math>f\!</math>
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| <math>f\!</math>
−
| style="width:16%" | <math>\operatorname{E}f|_{xy}\!</math>
+
| <math>\operatorname{E}f|_{xy}</math>
−
| style="width:16%" | <math>\operatorname{E}f|_{x(y)}\!</math>
+
| <math>\operatorname{E}f|_{x(y)}</math>
−
| style="width:16%" | <math>\operatorname{E}f|_{(x)y}\!</math>
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| <math>\operatorname{E}f|_{(x)y}</math>
−
| style="width:16%" | <math>\operatorname{E}f|_{(x)(y)}\!</math>
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| <math>\operatorname{E}f|_{(x)(y)}</math>
−
|-
+
|- style="height:36px"
 
| <math>f_{0}\!</math>
 
| <math>f_{0}\!</math>
 
| <math>(~)\!</math>
 
| <math>(~)\!</math>
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| <math>(~)\!</math>
 
| <math>(~)\!</math>
 
|-
 
|-
−
| <math>f_{1}\!</math>
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|
−
| <math>(x)(y)\!</math>
+
{| align="center"
−
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
+
|-
−
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
+
| height="36px" | <math>f_{1}\!</math>
−
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
+
|-
−
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
+
| height="36px" | <math>f_{2}\!</math>
 +
|-
 +
| height="36px" | <math>f_{4}\!</math>
 +
|-
 +
| height="36px" | <math>f_{8}\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(x)(y)\!</math>
 +
|-
 +
| height="36px" | <math>(x)\ y\!</math>
 +
|-
 +
| height="36px" | <math>x\ (y)\!</math>
 +
|-
 +
| height="36px" | <math>x\ y\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x) (\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)\ \operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ (\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x\ \operatorname{d}y</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>f_{3}\!</math>
 +
|-
 +
| height="36px" | <math>f_{12}\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(x)\!</math>
 +
|-
 +
| height="36px" | <math>x\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>\operatorname{d}x</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>\operatorname{d}x</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}x</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>f_{6}\!</math>
 +
|-
 +
| height="36px" | <math>f_{9}\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(x,\ y)\!</math>
 
|-
 
|-
−
| <math>f_{2}\!</math>
+
| height="36px" | <math>((x,\ y))\!</math>
−
| <math>(x) y\!</math>
+
|}
−
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
+
|
−
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
+
{| align="center"
−
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
  −
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
   
|-
 
|-
−
| <math>f_{4}\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
−
| <math>x (y)\!</math>
  −
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
  −
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
  −
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
  −
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
   
|-
 
|-
−
| <math>f_{8}\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
−
| <math>x y\!</math>
+
|}
−
| <math>(\operatorname{d}x)(\operatorname{d}y)\!</math>
+
|
−
| <math>(\operatorname{d}x) \operatorname{d}y\!</math>
+
{| align="center"
−
| <math>\operatorname{d}x (\operatorname{d}y)\!</math>
  −
| <math>\operatorname{d}x\ \operatorname{d}y\!</math>
   
|-
 
|-
−
| <math>f_{3}\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
−
| <math>(x)\!</math>
  −
| <math>\operatorname{d}x\!</math>
  −
| <math>\operatorname{d}x\!</math>
  −
| <math>(\operatorname{d}x)\!</math>
  −
| <math>(\operatorname{d}x)\!</math>
   
|-
 
|-
−
| <math>f_{12}\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
−
| <math>x\!</math>
+
|}
−
| <math>(\operatorname{d}x)\!</math>
+
|
−
| <math>(\operatorname{d}x)\!</math>
+
{| align="center"
−
| <math>\operatorname{d}x\!</math>
  −
| <math>\operatorname{d}x\!</math>
   
|-
 
|-
−
| <math>f_{6}\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
−
| <math>(x, y)\!</math>
+
|-
−
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
−
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
+
|}
−
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
+
|
−
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
+
{| align="center"
 
|-
 
|-
−
| <math>f_{9}\!</math>
+
| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</math>
−
| <math>((x, y))\!</math>
  −
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
  −
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
  −
| <math>(\operatorname{d}x, \operatorname{d}y)\!</math>
  −
| <math>((\operatorname{d}x, \operatorname{d}y))\!</math>
   
|-
 
|-
−
| <math>f_{5}\!</math>
+
| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
−
| <math>(y)\!</math>
+
|}
−
| <math>\operatorname{d}y\!</math>
  −
| <math>(\operatorname{d}y)\!</math>
  −
| <math>\operatorname{d}y\!</math>
  −
| <math>(\operatorname{d}y)\!</math>
   
|-
 
|-
−
| <math>f_{10}\!</math>
+
|
−
| <math>y\!</math>
+
{| align="center"
−
| <math>(\operatorname{d}y)\!</math>
  −
| <math>\operatorname{d}y\!</math>
  −
| <math>(\operatorname{d}y)\!</math>
  −
| <math>\operatorname{d}y\!</math>
   
|-
 
|-
−
| <math>f_{7}\!</math>
+
| height="36px" | <math>f_{5}\!</math>
−
| <math>(x y)\!</math>
  −
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
  −
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
  −
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
  −
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
   
|-
 
|-
−
| <math>f_{11}\!</math>
+
| height="36px" | <math>f_{10}\!</math>
−
| <math>(x (y))\!</math>
+
|}
−
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
+
|
−
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
+
{| align="center"
−
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
  −
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
   
|-
 
|-
−
| <math>f_{13}\!</math>
+
| height="36px" | <math>(y)\!</math>
−
| <math>((x) y)\!</math>
  −
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
  −
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
  −
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
  −
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
   
|-
 
|-
−
| <math>f_{14}\!</math>
+
| height="36px" | <math>y\!</math>
−
| <math>((x)(y))\!</math>
+
|}
−
| <math>(\operatorname{d}x\ \operatorname{d}y)\!</math>
+
|
−
| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
+
{| align="center"
−
| <math>((\operatorname{d}x) \operatorname{d}y)\!</math>
  −
| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
   
|-
 
|-
 +
| height="36px" | <math>\operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}y)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}y</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>\operatorname{d}y</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}y)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>\operatorname{d}y</math>
 +
|}
 +
|-
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>f_{7}\!</math>
 +
|-
 +
| height="36px" | <math>f_{11}\!</math>
 +
|-
 +
| height="36px" | <math>f_{13}\!</math>
 +
|-
 +
| height="36px" | <math>f_{14}\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(x\ y)\!</math>
 +
|-
 +
| height="36px" | <math>(x\ (y))\!</math>
 +
|-
 +
| height="36px" | <math>((x)\ y)\!</math>
 +
|-
 +
| height="36px" | <math>((x)(y))\!</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 +
|}
 +
|
 +
{| align="center"
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ \operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>(\operatorname{d}x\ (\operatorname{d}y))</math>
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)\ \operatorname{d}y)</math>
 +
|-
 +
| height="36px" | <math>((\operatorname{d}x)(\operatorname{d}y))</math>
 +
|}
 +
|- style="height:36px"
 
| <math>f_{15}\!</math>
 
| <math>f_{15}\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
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| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
 
| <math>((~))\!</math>
−
|}<br>
+
|}
 +
<br>
    
<pre>
 
<pre>
Jon Awbrey
12,080

edits

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