ONT Re: Reductions Among Relations
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RAR. Note 10
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Compositional Analysis of Relations (cont.)
Starting from the simplest notions of Rise and Fall,
I may easily have chosen to leave it as an exercise
for the reader to discover suitable generalizations,
say from Rise(k) and Fall(k) for k of order 2 and 3,
to the slightly more general case in which k is any
natural number, that is, finite, integral, positive.
But that is far too easy a calisthenic, and no kind
of a work-out to offer our band of fearless readers,
and so the writer picks up the gage that he himself
throws down, and for his health runs the easy track!
Let B = {0, 1}.
Let the relation Rise(k) c B^k
be defined in the following way:
| Rise(k)<x_1, ..., x_k>
|
| iff
|
| Rise(2)<x_1, x_2> and Rise(k-1)<x_2, ..., x_k>.
Let the relation Fall(k) c B^k
be defined in the following way:
| Fall(k)<x_1, ..., x_k>
|
| iff
|
| Fall(2)<x_1, x_2> and Fall(k-1)<x_2, ..., x_k>.
But let me now leave off, for the time being,
from the temptation to go any further in the
direction of increasing k than I ever really
intended to, on beyond 2 or 3 or thereabouts,
for that is not the aim of the present study.
Jon Awbrey
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