ONT Re: Extension x Comprehension = Information
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We have been contemplateing some of the more facile relationships
among functions, their fibers, and their factorizations. Just to
be as clear about all this as we possibly can, let us look at one
very simple but perfectly generic picture of the global situation.
The next Figure illustrates a function f : X -> Y with this data:
X = {x_1, x_2, x_3, x_4, x_5}
Y = {y_1, y_2, y_3, y_4, y_5, y_6}
f = {<x_1, y_2>,
<x_2, y_2>,
<x_3, y_2>,
<x_4, y_5>,
<x_5, y_5>}
o-----------------------------o-----------------------------o
| Source Domain | Target Codomain |
o-----------------------------o-----------------------------o
| |
| x_1 o--------------------------· o y_1 |
| \ |
| \ |
| x_2 o-----------------------------o y_2 |
| / |
| / |
| x_3 o--------------------------· o y_3 |
| |
| |
| x_4 o--------------------------· o y_4 |
| \ |
| \ |
| o y_5 |
| / |
| / |
| x_5 o--------------------------· o y_6 |
| |
| |
| Functional Fibers |
o-----------------------------------------------------------o
Just by way of introducing a few bits of useful terminology,
I take the liberty of expressing the following observations:
Dom(f) = Domain(f) = X
Cod(f) = Codomain(f) = Y
Ran(f) = Range(f) = {y_2, y_5}
Cor(f) = Corange(f) = X
Naturally, Dom(f) = Cor(f) for any relation f that
happens to be a function, but I am introducing these
terms as employed in a more general relational context.
The fibers of f are either one of these constructions:
1. Relational Fibers:
f & y_2 = {<x_1, y_2>,
<x_2, y_2>,
<x_3, y_2>}
f & y_5 = {<x_4, y_5>,
<x_5, y_5>}
2. Partitional Fibers:
f · y_2 = {x_1, x_2, x_3}
f · y_5 = {x_4, x_5}
There are, of course, many different systems
of notation and terminology for these things.
Jon Awbrey
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