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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 12:14, 10 February 2009

4,969 bytes added ,  12:14, 10 February 2009
→‎Fact 2.1
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} :D12a
 
} :D12a
 
</pre>
 
</pre>
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<br>
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{| align="center" cellpadding="0" 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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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:50px; text-align:center"
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| style="width:80%" | &nbsp;
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| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~1}</math>
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:50px"
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| style="border-top:1px solid black; width:2%"  | &nbsp;
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| style="border-top:1px solid black; width:18%" | <math>\text{If}\!</math>
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| style="border-top:1px solid black; width:60%" | <math>P, Q ~\subseteq~ X</math>
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| style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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|- style="height:50px"
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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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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:60px"
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| style="width:2%;  border-top:1px solid black" | &nbsp;
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| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>
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| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math>
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| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1a~:~R9a}</math>
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|- style="height:20px"
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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>
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|- style="height:40px"
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| &nbsp;
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| <math>\operatorname{F1b.}</math>
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| <math>s \quad \Leftrightarrow \quad (\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright)</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1b~:~R9b}</math>
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|- style="height:20px"
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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>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F1c.}</math>
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| <math>s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1c~:~R9c}</math>
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|- style="height:20px"
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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>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F1d.}</math>
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| <math>s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1d~:~R9d}</math></p>
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<p><math>\operatorname{F1d~:~R8a}</math></p>
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|- style="height:20px"
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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>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F1e.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1e~:~R8b}</math></p>
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<p><math>\operatorname{F1e~:~\_\_?\_\_}</math></p>
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|- style="height:20px"
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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>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F1f.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
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|- style="height:20px"
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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>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F1g.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
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|- style="height:20px"
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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>
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|- style="height:40px"
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| &nbsp;
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| <math>\operatorname{F1h.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1h~:~~\_\_?\_\_}</math>
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|}
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|}
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<br>
    
===Fact 2.2===
 
===Fact 2.2===
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/78708"