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Figure 3.2b
shows a many-to-one function). In a graph that is not a function
(not “single-valued”), the vertical line may cut this curve (that is not the graph
of a function) in more than one place (
Figure 3.2c
shows such a graph).
The visual display of the difference between function and relation, the many-
to-one and the one-to-many, is clear in the Cartesian coordinate system
because the ordering of the function from
X
to
Y
is clear in our minds. In a
coordinate-free environment, such as the world of the applet (a small applica-
tion that runs tasks within a larger program), all that is evident is the struc-
tural equivalence of many-to-one and one-to-many transformations. In
Figure
3.3,
note the stability of the one-to-one transformation as the graphic moves;
the many-to-one and the one-to-many never quite settle down to a totally sta-
ble configuration. This lack is a function of pattern involving length of edges
joining nodes and dimension of the square universe of discourse in which the
applets live. In the case of
Figure 3.3,
it may simply be a function of a par-
ticular commensurability pattern of edges and underlying raster; nonetheless,
the general consideration as to what sorts of configurations exhibit geometric
stability is an important one, particularly as in regard for looking for points
of intervention into process.
The relation is often ignored in mathematical analyses of various sorts. Perhaps
that is because the definite nature of single-valued mappings is regarded as
important. Is the world, however, single-valued? We consider a few real-world
Figure 3.3 Applets show one-to-one, many-to-one, and one-to-many transforma-
tions. Note the structural equivalence between the many-to-one and the one-
to-many applets. The printed image is a screen capture of the dynamic applet.
Source: Arlinghaus, S. L. and W. C. Arlinghaus. 2001. The Neglected Relation.
Solstice: An Electronic Journal of Geography and Mathematics. Vol. XII, No. 1.
edu/%7Ecopyrght/image/solstice/sum01/compplets.html
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