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− | <pre> | + | {| 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%" |
− | Geometric Translation Rule 2 | + | | |
− | | + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
− | If X, Y c U | + | |- style="height:48px; text-align:right" |
− | | + | | width="98%" | <math>\text{Geometric Translation Rule 2}\!</math> |
− | and P, Q U -> B, such that: | + | | width="2%" | |
− | | + | |} |
− | G2a. {X} = P and {Y} = Q, | + | |- |
− | | + | | |
− | then the following equations hold: | + | {| align="center" cellpadding="0" cellspacing="0" width="100%" |
− | | + | |- style="height:48px" |
− | G2b00. {{}} = () = 0 : U->B.
| + | | width="2%" style="border-top:1px solid black" | |
− | | + | | width="14%" style="border-top:1px solid black" | <math>\text{If}\!</math> |
− | G2b01. {~X n ~Y} = ({X})({Y}) = (P)(Q).
| + | | width="84%" style="border-top:1px solid black" | <math>P, Q \subseteq X</math> |
− | | + | |- style="height:48px" |
− | G2b02. {~X n Y} = ({X}){Y} = (P) Q.
| + | | |
− | | + | | <math>\text{and}\!</math> |
− | G2b03. {~X} = ({X}) = (P).
| + | | <math>p, q ~:~ X \to \underline\mathbb{B}</math> |
− | | + | |- style="height:48px" |
− | G2b04. {X n ~Y} = {X}({Y}) = P (Q).
| + | | |
− | | + | | <math>\text{such that:}\!</math> |
− | G2b05. {~Y} = ({Y}) = (Q).
| + | | |
− | | + | |- style="height:48px" |
− | G2b06. {X + Y} = ({X}, {Y}) = (P, Q).
| + | | |
− | | + | | <math>\text{G2a.}\!</math> |
− | G2b07. {~(X n Y)} = ({X}.{Y}) = (P Q).
| + | | <math>\upharpoonleft P \upharpoonright ~=~ p \quad \operatorname{and} \quad \upharpoonleft Q \upharpoonright ~=~ q</math> |
− | | + | |- style="height:48px" |
− | G2b08. {X n Y} = {X}.{Y} = P.Q.
| + | | |
− | | + | | <math>\text{then}\!</math> |
− | G2b09. {~(X + Y)} = (({X}, {Y})) = ((P, Q)).
| + | | <math>\text{the following equations hold:}\!</math> |
− | | + | |} |
− | G2b10. {Y} = {Y} = Q.
| + | |- |
− | | + | | |
− | G2b11. {~(X n ~Y)} = ({X}({Y})) = (P (Q)).
| + | {| align="center" cellpadding="0" cellspacing="0" style="text-align:center" width="100%" |
− | | + | |- style="height:52px" |
− | G2b12. {X} = {X} = P.
| + | | width="2%" style="border-top:1px solid black" | |
− | | + | | width="14%" style="border-top:1px solid black" align="left" | <math>\text{G2b}_{0}.\!</math> |
− | G2b13. {~(~X n Y)} = (({X}) {Y}) = ((P) Q).
| + | | width="32%" style="border-top:1px solid black" | |
− | | + | <math>\upharpoonleft \varnothing \upharpoonright</math> |
− | G2b14. {X u Y} = (({X})({Y})) = ((P)(Q)).
| + | | width="4%" style="border-top:1px solid black" | <math>=\!</math> |
− | | + | | width="28%" style="border-top:1px solid black" | <math>(~)</math> |
− | G2b15. {U} = (()) = 1 : U->B.
| + | | width="4%" style="border-top:1px solid black" | <math>=\!</math> |
− | </pre> | + | | width="16%" style="border-top:1px solid black" | <math>(~)</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{1}.\!</math> |
| + | | <math>\upharpoonleft \overline{P} ~\cap~ \overline{Q} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>(p)(q)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{2}.\!</math> |
| + | | <math>\upharpoonleft \overline{P} ~\cap~ Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(p) q\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{3}.\!</math> |
| + | | <math>\upharpoonleft \overline{P} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft P \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>(p)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{4}.\!</math> |
| + | | <math>\upharpoonleft P ~\cap~ \overline{Q} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>p (q)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{5}.\!</math> |
| + | | <math>\upharpoonleft \overline{Q} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft Q \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>(q)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{6}.\!</math> |
| + | | <math>\upharpoonleft P ~+~ Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>(p, q)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{7}.\!</math> |
| + | | <math>\upharpoonleft \overline{P ~\cap~ Q} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>(p q)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{8}.\!</math> |
| + | | <math>\upharpoonleft P ~\cap~ Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>\upharpoonleft P \upharpoonright ~ \upharpoonleft Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>p q\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{9}.\!</math> |
| + | | <math>\upharpoonleft \overline{P ~+~ Q} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>((\upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright))</math> |
| + | | <math>=\!</math> |
| + | | <math>((p, q))\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{10}.\!</math> |
| + | | <math>\upharpoonleft Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>\upharpoonleft Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>q\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{11}.\!</math> |
| + | | <math>\upharpoonleft \overline{P ~\cap~ \overline{Q}} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>(\upharpoonleft P \upharpoonright (\upharpoonleft Q \upharpoonright))</math> |
| + | | <math>=\!</math> |
| + | | <math>(p (q))\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{12}.\!</math> |
| + | | <math>\upharpoonleft P \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>\upharpoonleft P \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>p\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{13}.\!</math> |
| + | | <math>\upharpoonleft \overline{\overline{P} ~\cap~ Q} \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>((\upharpoonleft P \upharpoonright) \upharpoonleft Q \upharpoonright)</math> |
| + | | <math>=\!</math> |
| + | | <math>((p) q)\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{14}.\!</math> |
| + | | <math>\upharpoonleft P ~\cup~ Q \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>((\upharpoonleft P \upharpoonright)(\upharpoonleft Q \upharpoonright))</math> |
| + | | <math>=\!</math> |
| + | | <math>((p)(q))\!</math> |
| + | |- style="height:52px" |
| + | | |
| + | | align="left" | <math>\text{G2b}_{15}.\!</math> |
| + | | <math>\upharpoonleft X \upharpoonright</math> |
| + | | <math>=\!</math> |
| + | | <math>((~))</math> |
| + | | <math>=\!</math> |
| + | | <math>((~))</math> |
| + | |} |
| + | |} |
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