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Directory:Jon Awbrey/Papers/Syntactic Transformations (view source)
Revision as of 04:36, 3 February 2009
, 04:36, 3 February 2009→Syntactic Transformation Rules: format rule box
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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%"
Rule 5
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If X, Y c U,
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then the following are equivalent:
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| width="20%" style="border-left:1px solid black" | <math>\operatorname{Rule~5}</math>
R5a. X = Y. :D2a
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R5b. u C X <=> u C Y, for all u C U. :D2b
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R5c. [u C X] = [u C Y], for all u C U. :D7b
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| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
| width="60%" style="border-top:1px solid black" | <math>P, Q ~\subseteq~ X</math>
R5d. {<u, v> C UxB : v = [u C X]}
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| <math>\text{then}\!</math>
| <math>\text{the following are equivalent:}\!</math>
R5e. {X} = {Y}. :D5a
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</pre>
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| width="18%" style="border-top:1px solid black" | <math>\operatorname{R5a.}</math>
| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R5a~:~D2a}</math>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R5b.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)</math>
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<p><math>\operatorname{R5b~:~D2b}</math></p>
<p><math>\operatorname{R5b~:~D7a}</math></p>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R5c.}</math>
| <math>\overset{X}{\underset{x}{\forall}}~ (\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright)</math>
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<p><math>\operatorname{R5c~:~D7b}</math></p>
<p><math>\operatorname{R5c~:~\_\_?\_\_}</math></p>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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| <math>\operatorname{R5d.}</math>
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<math>\begin{matrix}
\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in P \downharpoonright
\\
=
\\
\{ (x, y) \in X \times \underline\mathbb{B} ~:~ y ~=~ \downharpoonleft x \in Q \downharpoonright
\end{matrix}</math>
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<p><math>\operatorname{R5d~:~\_\_?\_\_}</math></p>
<p><math>\operatorname{R5d~:~D5b}</math></p>
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
|- style="height:40px"
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| <math>\operatorname{R5e.}</math>
| <math>\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright</math>
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R5e~:~D5a}</math>
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