Changes

User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 20:02, 31 January 2009

689 bytes added , 20:02, 31 January 2009

→‎Rule 5

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===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>~~

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|- style="height:40px; text-align:center"

| width="80%" |  

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| 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"

| width="2%" style="border-top:1px solid black" |  

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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" |  

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~~|- style="height:40px"~~

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~~|  ~~

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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" |  ~~

|- style="height:40px"

|  

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|- style="height:40px"

| width="2%" style="border-top:1px solid black" |  

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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"

|  

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|- style="height:60px"

|  

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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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<math>\operatorname{R5b~:~D2b}</math>

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<math>\operatorname{R5b~:~D7a}</math>

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|- style="height:20px"

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|  

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|  

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|  

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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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|  

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| <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>

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| style="border-left:1px solid black; text-align:center" |

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<math>\operatorname{R5c~:~D7b}</math>

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<math>\operatorname{R5c~:~\_\_?\_\_}</math>

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|- style="height:20px"

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|  

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|  

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|  

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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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|  

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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" |

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<math>\operatorname{~~R3b~~~:~~~R1b~~}</math>

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<math>\operatorname{R5d~:~\_\_?\_\_}</math>

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<math>\operatorname{~~R3b~~~:~~~R2a~~}</math>

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<math>\operatorname{R5d~:~D5b}</math>

|- style="height:20px"

|  

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|- style="height:40px"

|  

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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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