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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 02:14, 29 January 2009

902 bytes removed ,  02:14, 29 January 2009
→‎Geometric Translation Rule 2
Line 1,378: Line 1,378:     
===Geometric Translation Rule 2===
 
===Geometric Translation Rule 2===
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<pre>
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Geometric Translation Rule 2
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If X, Y c U
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and P, Q U -> B, such that:
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G2a. {X} = P  and  {Y} = Q,
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then the following equations hold:
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G2b00. {{}} = () = 0 : U->B.
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G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
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G2b02. {~X n Y} = ({X}){Y} = (P) Q.
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G2b03. {~X} = ({X}) = (P).
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G2b04. {X n ~Y} = {X}({Y}) = P (Q).
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G2b05. {~Y} = ({Y}) = (Q).
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G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
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G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
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G2b08. {X n Y} = {X}.{Y} = P.Q.
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G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
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G2b10. {Y} = {Y} = Q.
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G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
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G2b12. {X} = {X} = P.
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G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
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G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
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G2b15. {U} = (()) = 1 : U->B.
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</pre>
      
<br>
 
<br>
Line 1,471: Line 1,427:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{1}.\!</math>
 
| align="left" | <math>\text{G2b}_{1}.\!</math>
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| <math>\upharpoonleft {}^{_\sim} P ~\cap~ {}^{_\sim} Q \upharpoonright</math>
+
| <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft {}^{_\sim} P \upharpoonright)(\upharpoonleft {}^{_\sim} Q \upharpoonright)</math>
+
| <math>(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(p)(q)\!</math>
 
| <math>(p)(q)\!</math>
Line 1,479: Line 1,435:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{2}.\!</math>
 
| align="left" | <math>\text{G2b}_{2}.\!</math>
−
| <math>\upharpoonleft \operatorname{not}~ s ~\operatorname{but}~ t \upharpoonright</math>
+
| <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft s \upharpoonright) \upharpoonleft t \upharpoonright</math>
+
| <math>(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(p) q\!</math>
 
| <math>(p) q\!</math>
Line 1,487: Line 1,443:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{3}.\!</math>
 
| align="left" | <math>\text{G2b}_{3}.\!</math>
−
| <math>\upharpoonleft \operatorname{not}~ s \upharpoonright</math>
+
| <math>\upharpoonleft \overline{P} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft s \upharpoonright)</math>
+
| <math>(\upharpoonleft P \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(p)\!</math>
 
| <math>(p)\!</math>
Line 1,495: Line 1,451:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{4}.\!</math>
 
| align="left" | <math>\text{G2b}_{4}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{and~not}~ t \upharpoonright</math>
+
| <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\upharpoonleft s \upharpoonright (\upharpoonleft t \upharpoonright)</math>
+
| <math>\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>p (q)\!</math>
 
| <math>p (q)\!</math>
Line 1,503: Line 1,459:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{5}.\!</math>
 
| align="left" | <math>\text{G2b}_{5}.\!</math>
−
| <math>\upharpoonleft \operatorname{not}~ t \upharpoonright</math>
+
| <math>\upharpoonleft \overline{Q} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft t \upharpoonright)</math>
+
| <math>(\upharpoonleft Q \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(q)\!</math>
 
| <math>(q)\!</math>
Line 1,511: Line 1,467:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{6}.\!</math>
 
| align="left" | <math>\text{G2b}_{6}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{or}~ t, ~\operatorname{not~both} \upharpoonright</math>
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| <math>\upharpoonleft P ~+~ Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft s \upharpoonright ~,~ \upharpoonleft t \upharpoonright)</math>
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| <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(p, q)\!</math>
 
| <math>(p, q)\!</math>
Line 1,519: Line 1,475:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{7}.\!</math>
 
| align="left" | <math>\text{G2b}_{7}.\!</math>
−
| <math>\upharpoonleft \operatorname{not~both}~ s ~\operatorname{and}~ t \upharpoonright</math>
+
| <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft s \upharpoonright ~ \upharpoonleft t \upharpoonright)</math>
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| <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(p q)\!</math>
 
| <math>(p q)\!</math>
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| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{8}.\!</math>
 
| align="left" | <math>\text{G2b}_{8}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{and}~ t \upharpoonright</math>
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| <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\upharpoonleft s \upharpoonright ~ \upharpoonleft t \upharpoonright</math>
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| <math>\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>p q\!</math>
 
| <math>p q\!</math>
Line 1,535: Line 1,491:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{9}.\!</math>
 
| align="left" | <math>\text{G2b}_{9}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{is~equivalent~to}~ t \upharpoonright</math>
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| <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>((\upharpoonleft s \upharpoonright ~,~ \upharpoonleft t \upharpoonright))</math>
+
| <math>((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>((p, q))\!</math>
 
| <math>((p, q))\!</math>
Line 1,543: Line 1,499:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{10}.\!</math>
 
| align="left" | <math>\text{G2b}_{10}.\!</math>
−
| <math>\upharpoonleft t \upharpoonright</math>
+
| <math>\upharpoonleft Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\upharpoonleft t \upharpoonright</math>
+
| <math>\upharpoonleft Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>q\!</math>
 
| <math>q\!</math>
Line 1,551: Line 1,507:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{11}.\!</math>
 
| align="left" | <math>\text{G2b}_{11}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{implies}~ t \upharpoonright</math>
+
| <math>\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>(\upharpoonleft s \upharpoonright (\upharpoonleft t \upharpoonright))</math>
+
| <math>(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>(p (q))\!</math>
 
| <math>(p (q))\!</math>
Line 1,559: Line 1,515:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{12}.\!</math>
 
| align="left" | <math>\text{G2b}_{12}.\!</math>
−
| <math>\upharpoonleft s \upharpoonright</math>
+
| <math>\upharpoonleft P \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>\upharpoonleft s \upharpoonright</math>
+
| <math>\upharpoonleft P \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>p\!</math>
 
| <math>p\!</math>
Line 1,567: Line 1,523:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{13}.\!</math>
 
| align="left" | <math>\text{G2b}_{13}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{is~implied~by}~ t \upharpoonright</math>
+
| <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>((\upharpoonleft s \upharpoonright) \upharpoonleft t \upharpoonright)</math>
+
| <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>((p) q)\!</math>
 
| <math>((p) q)\!</math>
Line 1,575: Line 1,531:  
| &nbsp;
 
| &nbsp;
 
| align="left" | <math>\text{G2b}_{14}.\!</math>
 
| align="left" | <math>\text{G2b}_{14}.\!</math>
−
| <math>\upharpoonleft s ~\operatorname{or}~ t \upharpoonright</math>
+
| <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
−
| <math>((\upharpoonleft s \upharpoonright)(\upharpoonleft t \upharpoonright))</math>
+
| <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</math>
 
| <math>=\!</math>
 
| <math>=\!</math>
 
| <math>((p)(q))\!</math>
 
| <math>((p)(q))\!</math>
Jon Awbrey
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