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Revision as of 06:22, 31 May 2008
, 06:22, 31 May 2008→Propositional Forms on Two Variables: sub new table
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| style="width:16%" | +| style="width:16%" | <math>f\!</math>+| style="width:16%" | <math>\operatorname{E}f|_{xy}\!</math>+| style="width:16%" | <math>\operatorname{E}f|_{x(y)}\!</math>+| style="width:16%" | <math>\operatorname{E}f|_{(x)y}\!</math>+| style="width:16%" | <math>\operatorname{E}f|_{(x)(y)}\!</math>+| <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>x y\!</math>+| <math>\operatorname{d}x\!</math>
−| <math>\operatorname{d}x\!</math>
−| <math>(\operatorname{d}x)\!</math>
−| <math>(\operatorname{d}x)\!</math>
| <math>x\!</math>+| <math>(\operatorname{d}x)\!</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>((\operatorname{d}x, \operatorname{d}y))\!</math>
| <math>\operatorname{d}y\!</math>
−| <math>(\operatorname{d}y)\!</math>
−| <math>\operatorname{d}y\!</math>
−| <math>y\!</math>+| <math>(\operatorname{d}y)\!</math>
−| <math>\operatorname{d}y\!</math>
−| <math>(\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>((\operatorname{d}x) \operatorname{d}y)\!</math>+| <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>
−
| <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>
+| <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"
−|
−| <math>f\!</math>
−| <math>\operatorname{E}f|_{xy}</math>
−| <math>\operatorname{E}f|_{x(y)}</math>
−| <math>\operatorname{E}f|_{(x)y}</math>
−| <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>
+|
−| <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>
+{| 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>(\operatorname{d}x)(\operatorname{d}y)\!</math>
−|-
|-
−| <math>f_{8}\!</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>
+{| 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>f_{12}\!</math>
+| height="36px" | <math>(\operatorname{d}x,\ \operatorname{d}y)</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>
+|-
−| 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>f_{5}\!</math>
+| height="36px" | <math>((\operatorname{d}x,\ \operatorname{d}y))</math>
−| <math>(y)\!</math>
+|}
−| <math>(\operatorname{d}y)\!</math>
|-
|-
−| <math>f_{10}\!</math>
+|
−{| align="center"
−| <math>\operatorname{d}y\!</math>
|-
|-
−| <math>f_{7}\!</math>
+| height="36px" | <math>f_{5}\!</math>
−|-
|-
−| <math>f_{11}\!</math>
+| height="36px" | <math>f_{10}\!</math>
−| <math>(x (y))\!</math>
+|}
−|
−| <math>((\operatorname{d}x)(\operatorname{d}y))\!</math>
+{| align="center"
−| <math>(\operatorname{d}x (\operatorname{d}y))\!</math>
|-
|-
−| <math>f_{13}\!</math>
+| height="36px" | <math>(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>
|-
|-
+| 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>
