Hitoshi Kitada (hitoshi@kitada.com)
Sun, 11 Apr 1999 08:24:52 +0900
Dear Stephen,
Just on the following point:
> > E.g., consider a set L={1,2,3}. (In the case of the universe, L may be an
> > infinite set. At this point, to use the notion "cluster decomposition" b
> > concerning the universe may be an abuse at least at the present stage of
the
> > theory. This point might be related with Obler's paradox as I mention
below.)
> >
> > Then the set Q is
> >
> > Q={ {{1},{2},{3}},
> > {{1},{2,3}}, {{1,2},{3}}, {{1,3},{2}},
> > {{1,2,3}} },
> >
> > consisting of 5 elements. b varies over those elements.
>
> Forgive my mathematical naivete, but is Q here an example of a power
> set?
The power set P of L={1,2,3} is
P={ {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
that is different from Q. Q is the set of decomposition b of the set L. A
decomposition b of L is a set defined as
b={C1,C2,...,Ck},
where Cj is a subset of L, which are mutually disjoint (i.e. Ci and Cj do not
intersect when i is not equal to j), and the sum set of C1,...,Ck is equal to
L. E.g.,
b={{1,2}, {3}}, etc.
Best wishes,
Hitoshi
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