ONT Re: Differential Logic
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DLOG. Note D73
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Transformations of Type B^2 -> B^2
To take up a slightly more complex example, but one that remains
simple enough to pursue through a complete series of developments,
consider the transformation from U% = [u, v] to X% = [x, y] that is
defined by the following system of equations:
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| |
| x = f(u, v) = ((u)(v)) |
| |
| y = g(u, v) = ((u, v)) |
| |
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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 define it by means of the compact description that follows:
o-----------------------------------------------------------o
| |
| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| |
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The information that defines the logical transformation F can be represented
in the form of a truth table, as in Table 60. To cut down on subscripts in
this example I continue to use plain letter equivalents for all components
of spaces and maps.
Table 60. A Propositional Transformation
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| u | v | f | g |
o-------------o-------------o-------------o-------------o
| | | | |
| 0 | 0 | 0 | 1 |
| | | | |
| 0 | 1 | 1 | 0 |
| | | | |
| 1 | 0 | 1 | 0 |
| | | | |
| 1 | 1 | 1 | 1 |
| | | | |
o-------------o-------------o-------------o-------------o
| | | ((u)(v)) | ((u, v)) |
o-------------o-------------o-------------o-------------o
Jon Awbrey
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