ONT Re: Theory Of Relations
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TOR. Note 5
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In as much as relations are nothing but aggregates, sets, or logical sums
of elementary relations (or ordered tuples), we may do with relations the
whole variety of familiar set-theoretic operations that we are accustomed
to carry out with the category of sets from which relations inherit their
properties as sets, for instance: complementation, intersection, union,
asymmetric difference, symmetric difference, and all the rest.
As a general rule, especially in the earliest papers, Peirce will analogize
intersections with products, unions with sums, whose sigils are !P! and !S!,
respectively, and he also dubs them the "non-relative aggregate or sum" and
the "non-relative composite or product", respectively, as best I can recall.
By way of acquiring some practical experience with the materials and tools in
this shop, let us devise a concrete example, whose study should be sufficient.
| Example 1.
|
| A = i:j + j:k + k:i
|
| B = i:k + j:i + k:j
|
| L = i:i + i:j + i:k + j:i + j:j + j:k + k:i + k:j + k:k
For the immediate if not exactly the unmitigated future,
we will contemplate only those sorts of operations that
are defined on relations of the same arities or types.
It is possible to get fancier about this, speaking of
formal objects called "relation complexes", but then
we would have to be very careful about what we mean
by expressions like "i + i:j + i:j:k", and how the
sets !L!_1, !L!_2, !L!_3, ... "embed" or "inject"
themselves into a more encompassing family !L!.
I judge that this'd be too distracting at this
stage of the game, so let's not go there, yet.
| Example 1.
|
| 1. The complement of A in L.
|
| ~A = L - A = i:i + i:k + j:i + j:j + k:j + k:k
|
| 2. The complement of B in L.
|
| ~B = L - B = i:i + i:j + j:j + j:k + k:i + k:k
|
| 3. The intersection or non-relative product of A and B.
|
| A |^| B = {} = the empty set, so A and B are "disjoint".
|
| 4. The union or non-relative sum of A and B.
|
| A |_| B = A + B = i:j + j:k + k:i + i:k + j:i + k:j
|
| 5. Since A and B are disjoint, we have the following facts
| about their differences and their symmetric difference:
|
| A - B = A
| B - A = B
| A ± B = A + B = A |_| B
I am writing this from memories in deep cold-storage.
Modulo the factors of memory and fallibility I think
that this is more or less how it goes but I may need
to go back and retrieve Peirce's actual notations at
some point.
Jon Awbrey
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