ONT Re: Reductions Among Relations
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RAR. Note 11
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Compositional Analysis of Relations (cont.)
In this note I revisit the "Between" relation on reals,
and then I rework it as a discrete and finite analogue
of its transcendantal self, as a Between relation on B.
Ultimately, I want to use this construction as working
material to illustrate a method of defining relational
compositions in terms of projections. So let us begin.
Last time I defined Rise and Fall relations on B^k.
Working polymorphously, as some people like to say,
let us go ahead and define the analogous relations
over the real domain R, not even bothering to make
new names, but merely expecting the reader to find
the aptest sense for a given context of discussion.
Let R be the set of real numbers.
Let the relation named "Rise(2)"
such that Rise(2) c R^2 = R x R,
be defined in the following way:
| Rise(2)<x, y>
|
| iff
|
| [x = y] or [x < y]
Let the relation named "Fall(2)"
such that Fall(2) c R^2 = R x R,
be defined in the following way:
| Fall(2)<x, y>
|
| iff
|
| [x > y] or [x = y]
There are clearly a number of redundancies
between the definitions of these relations,
but I prefer the symmetry of this approach.
The next pair of definitions will be otiose, too,
if viewed in the light of the comprehensive case
that follows after, but let us go gently for now.
Let the relation named "Rise(3)"
such that Rise(3) c R^3 = RxRxR,
be defined in the following way:
| Rise(3)<x, y, z>
|
| iff
|
| Rise(2)<x, y> and Rise(2)<y, z>
Let the relation named "Fall(3)"
such that Fall(3) c R^3 = RxRxR,
be defined in the following way:
| Fall(3)<x, y, z>
|
| iff
|
| Fall(2)<x, y> and Fall(2)<y, z>
Then Rise(3) and Fall(3) are "degenerate 3-adic relations"
insofar as each of them bears expression as a conjunction
whose conjuncts are expressions of 2-adic relations alone.
Just in order to complete the development
of this thought, let us then finish it so:
Let the relation Rise(k) c R^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 R^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>
If there was a point to writing out this last step,
I think that it may well have been how easy it was
not to write, not literally to "write" at all, but
simply to "cut and paste" the definitions from the
boolean case, and then but to change the parameter
B into the parameter R at a mere one place in each.
Jon Awbrey
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