Changes

User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 17:30, 2 February 2009

1,751 bytes added , 17:30, 2 February 2009

→‎Rule 11

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===Rule 11===

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~~<pre>~~

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~~Rule 11~~

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~~If X c U~~

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~~then the following are equivalent:~~

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~~R11a. X = {u C U : S}. :R5a~~

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::

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~~R11b. {X} = { {u C U : S} }. :R5e~~

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::

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~~R11c. {X} c UxB~~

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~~: {X} = {<u, v> C UxB : v = [S](u)}. :R~~

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::

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~~R11d. {X} : U -> B~~

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~~: {X}(u) = [S](u), for all u C U. :R~~

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::

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~~R11e. {X} = [S]. :R~~

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~~</pre>~~

====Variant 1====

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

| width="80%" |  

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| width="20%" | <math>\~~text~~{~~Definition 5~~}\!</math>

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| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~11}</math>

|}

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

| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>

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| width="80%" style="border-top:1px solid black" | <math>Q ~\subseteq~ X</math>

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| width="60%" style="border-top:1px solid black" | <math>Q ~\subseteq~ X</math>

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| width="20%" style="border-top:1px solid black; border-left:1px solid black" |  

|- style="height:40px"

|  

| <math>\text{then}\!</math>

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| <math>\text{the following are ~~identical propositions~~:}\!</math>

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| <math>\text{the following are equivalent:}\!</math>

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| style="border-left:1px solid black" |  

|}

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|- style="height:40px"

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

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| width="18%" style="border-top:1px solid black" | <math>\operatorname{~~D5a~~.}</math>

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| width="18%" style="border-top:1px solid black" | <math>\operatorname{R11a.}</math>

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| width="80%" style="border-top:1px solid black" | <math>\upharpoonleft Q \upharpoonright</math>

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| width="60%" style="border-top:1px solid black" | <math>Q ~=~ \{ x \in X ~:~ s \}</math>

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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R11a~:~R5a}</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:40px"

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|  

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| <math>\operatorname{R11b.}</math>

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| <math>\upharpoonleft Q \upharpoonright ~=~ \upharpoonleft \{ x \in X ~:~ s \} \upharpoonright</math>

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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R11b~:~R5e}</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{R11c.}</math>

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|

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<math>\begin{array}{lcl}

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\upharpoonleft Q \upharpoonright

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& \subseteq &

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X \times \underline\mathbb{B}

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\\

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\upharpoonleft Q \upharpoonright

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& = &

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\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y = \, \downharpoonleft s \downharpoonright (x) \}

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\end{array}</math>

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

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<math>\operatorname{R11c~:~\_\_?\_\_}</math>

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<math>\operatorname{R11c~:~\_\_?\_\_}</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>

|- style="height:60px"

|  

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| <math>\operatorname{~~D5b~~.}</math>

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| <math>\operatorname{R11d.}</math>

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<math>\begin{array}{ccccl}

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f & : & X & \to & \underline\mathbb{B}

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\upharpoonleft Q \upharpoonright & : & X & \to & \underline\mathbb{B}

\\

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f & : & x & \mapsto & \downharpoonleft ~~x \in Q~~ \downharpoonright

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\upharpoonleft Q \upharpoonright & : & x & \mapsto & \downharpoonleft s \downharpoonright (x)

\end{array}</math>

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

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<math>\operatorname{R11d~:~\_\_?\_\_}</math>

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<math>\operatorname{R11d~:~\_\_?\_\_}</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:40px"

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|  

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| <math>\operatorname{R11e.}</math>

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| <math>\upharpoonleft Q \upharpoonright ~=~ \downharpoonleft s \downharpoonright</math>

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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R11e~:~\_\_?\_\_}</math>

|}

Jon Awbrey

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