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| Onward and upward to Flatland, the differential analysis of transformations between 2-dimensional universes of discourse. | | Onward and upward to Flatland, the differential analysis of transformations between 2-dimensional universes of discourse. |
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− | <pre>
| + | Consider the transformation from the universe <math>U^\circ = [u, v]</math> to the universe <math>X^\circ = [x, y]</math> that is defined by this system of equations: |
− | Consider the transformation from the universe U% = [u, v] to the | |
− | universe X% = [x, y] that is defined by this system of equations: | |
| | | |
− | x = f<u, v> = ((u)(v)) | + | {| align="center" cellpadding="8" width="90%" |
| + | | <math>x ~=~ f(u, v) ~=~ \underline{((}~ u ~\underline{)(}~ v ~\underline{))}</math> |
| + | |- |
| + | | <math>y ~=~ g(u, v) ~=~ \underline{((}~ u ~,~ v ~\underline{))}</math> |
| + | |} |
| | | |
− | y = g<u, v> = ((u, v))
| + | The underlined parenthetical expressions on the right are the cactus forms for the boolean functions that correspond to inclusive disjunction and logical equivalence, respectively. By way of a reminder, consult Table 1 on the page at this location: |
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− | The parenthetical expressions on the right are the cactus forms for
| + | :* [http://stderr.org/pipermail/inquiry/2003-May/000478.html DLOG D1] |
− | the boolean functions that correspond to inclusive disjunction and
| |
− | logical equivalence, respectively. By way of a reminder, consult
| |
− | Table 1 on the page at this location:
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− | | |
− | DLOG D1. http://stderr.org/pipermail/inquiry/2003-May/000478.html
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| + | <pre> |
| The component notation F = <F_1, F_2> = <f, g> : U% -> X% allows | | The component notation F = <F_1, F_2> = <f, g> : U% -> X% allows |
| us to give a name and a type to this transformation, and permits | | us to give a name and a type to this transformation, and permits |