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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 21:34, 29 January 2009

1,142 bytes removed ,  21:34, 29 January 2009
→‎Variant 1
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| width="2%"  style="border-top:1px solid black" |  
 
| width="2%"  style="border-top:1px solid black" |  
 
| width="14%" style="border-top:1px solid black" | <math>\text{E1a.}\!</math>
 
| width="14%" style="border-top:1px solid black" | <math>\text{E1a.}\!</math>
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| width="84%" style="border-top:1px solid black" | <math>f(x) ~=~ g(x)</math>
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| width="64%" style="border-top:1px solid black" | <math>f(x) ~=~ g(x)</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\text{E1a : V1a}\!</math>
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|- style="height:24px"
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| &nbsp;
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| &nbsp;
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| &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
|- style="height:56px"
 
|- style="height:56px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\text{E1b.}\!</math>
 
| <math>\text{E1b.}\!</math>
 
| <math>f(x) ~\Leftrightarrow~ g(x)</math>
 
| <math>f(x) ~\Leftrightarrow~ g(x)</math>
 +
| style="border-left:1px solid black; text-align:center" | <math>\text{E1b : V1b}\!</math>
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|- style="height:24px"
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| &nbsp;
 +
| &nbsp;
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| &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
|- style="height:56px"
 
|- style="height:56px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\text{E1c.}\!</math>
 
| <math>\text{E1c.}\!</math>
 
| <math>\underline{((}~ f(x) ~,~ g(x) ~\underline{))}</math>
 
| <math>\underline{((}~ f(x) ~,~ g(x) ~\underline{))}</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\text{E1c : V1c}\!</math></p>
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<p><math>\text{E1c : S1a}\!</math></p>
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|- style="height:24px"
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| &nbsp;
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| &nbsp;
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| &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
|- style="height:56px"
 
|- style="height:56px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\text{E1d.}\!</math>
 
| <math>\text{E1d.}\!</math>
 
| <math>\underline{((}~ f ~,~ g ~\underline{))}^\$ (x)</math>
 
| <math>\underline{((}~ f ~,~ g ~\underline{))}^\$ (x)</math>
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| style="border-left:1px solid black; text-align:center" | <math>\text{E1d : S1b}\!</math>
 
|}
 
|}
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|}
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<br>
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{| align="center" cellpadding="2" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
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|- style="height:40px"
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| width="2%"  | &nbsp;
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| width="18%" | &nbsp;
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| width="60%" | &nbsp;
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| align="center" style="border-left:1px solid black" width="20%" | <math>\text{Rule 3}\!</math>
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|- style="height:40px"
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| style="border-top:1px solid black" | &nbsp;
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| style="border-top:1px solid black" | <math>\text{If}\!</math>
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| style="border-top:1px solid black" | <math>Q \subseteq X</math>
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| style="border-left:1px solid black; border-top:1px solid black" | &nbsp;
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|- style="height:40px"
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| &nbsp;
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| <math>\text{and}\!</math>
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| <math>x \in X</math>
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| style="border-left:1px solid black" | &nbsp;
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|- style="height:40px"
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| &nbsp;
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| <math>\text{then}\!</math>
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| <math>\text{the following are equivalent:}\!</math>
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| style="border-left:1px solid black" | &nbsp;
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|- style="height:36px"
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| style="border-top:1px solid black" | &nbsp;
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| style="border-top:1px solid black" | <math>\text{R3a.}\!</math>
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| style="border-top:1px solid black" | <math>x \in Q</math>
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| align="center" style="border-left:1px solid black; border-top:1px solid black" | <math>\text{R3a : R1a}\!</math>
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|- style="height:24px"
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| &nbsp; || &nbsp; || &nbsp;
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| align="center" style="border-left:1px solid black" | <math>::\!</math>
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|- style="height:36px"
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| &nbsp;
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| <math>\text{R3b.}\!</math>
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| <math>\upharpoonleft Q \upharpoonright (x)</math>
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| align="center" style="border-left:1px solid black" |
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<p><math>\text{R3b : R1b}\!</math></p>
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<p><math>\text{R3b : R2a}\!</math></p>
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|- style="height:24px"
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| &nbsp; || &nbsp; || &nbsp;
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| align="center" style="border-left:1px solid black" | <math>::\!</math>
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|- style="height:36px"
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| &nbsp;
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| <math>\text{R3c.}\!</math>
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| <math>\upharpoonleft Q \upharpoonright (x) = \underline{1}</math>
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| align="center" style="border-left:1px solid black" | <math>\text{R3c : R2b}\!</math>
   
|}
 
|}
  
Jon Awbrey
12,089

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