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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 20:02, 31 January 2009

689 bytes added ,  20:02, 31 January 2009
→‎Rule 5
Line 2,459: Line 2,459:     
===Rule 5===
 
===Rule 5===
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<pre>
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Rule 5
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If X, Y c U,
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then the following are equivalent:
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R5a. X = Y. :D2a
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::
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R5b. u C X  <=>  u C Y, for all u C U. :D2b
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:D7a
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::
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R5c. [u C X] = [u C Y], for all u C U. :D7b
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:???
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::
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R5d. {<u, v> C UxB : v = [u C X]}
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=
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{<u, v> C UxB : v = [u C Y]}. :???
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:D5b
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::
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R5e. {X} = {Y}. :D5a
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</pre>
      
<br>
 
<br>
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|- style="height:40px; text-align:center"
 
|- style="height:40px; text-align:center"
 
| width="80%" | &nbsp;
 
| width="80%" | &nbsp;
−
| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~3}</math>
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| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~5}</math>
 
|}
 
|}
 
|-
 
|-
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|- style="height:40px"
 
|- style="height:40px"
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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| width="64%" style="border-top:1px solid black" | <math>Q ~\subseteq~ X</math>
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| width="60%" style="border-top:1px solid black" | <math>P, Q ~\subseteq~ X</math>
 
| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
 
| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
−
|- 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;
   
|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:40px"
 
|- style="height:40px"
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
 
| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="14%" style="border-top:1px solid black" | <math>\operatorname{R3a.}</math>
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| width="18%" style="border-top:1px solid black" | <math>\operatorname{R5a.}</math>
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| width="64%" style="border-top:1px solid black" | <math>x ~\in~ Q</math>
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| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R3a~:~R1a}</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R5a~:~D2a}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:60px"
 
|- style="height:60px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{R3b.}</math>
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| <math>\operatorname{R5b.}</math>
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| <math>\upharpoonleft Q \upharpoonright (x)</math>
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| <math>\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)</math>
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| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{R5b~:~D2b}</math></p>
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<p><math>\operatorname{R5b~:~D7a}</math></p>
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|- style="height:20px"
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| &nbsp;
 +
| &nbsp;
 +
| &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;
 +
| <math>\operatorname{R5c.}</math>
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| <math>\overset{X}{\underset{x}{\forall}}~ (\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright)</math>
 +
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{R5c~:~D7b}</math></p>
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<p><math>\operatorname{R5c~:~\_\_?\_\_}</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:80px"
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| &nbsp;
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| <math>\operatorname{R5d.}</math>
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|
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<math>\begin{matrix}
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\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in P \downharpoonright
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\\
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=
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\\
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\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in Q \downharpoonright
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\end{matrix}</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{R3b~:~R1b}</math></p>
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<p><math>\operatorname{R5d~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{R3b~:~R2a}</math></p>
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<p><math>\operatorname{R5d~:~D5b}</math></p>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{R3c.}</math>
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| <math>\operatorname{R5e.}</math>
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| <math>\upharpoonleft Q \upharpoonright (x) ~=~ \underline{1}</math>
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| <math>\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R3c~:~R2b}</math></p>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R5e~:~D5a}</math>
 
|}
 
|}
 
|}
 
|}
Jon Awbrey
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