Second Inevitable Correction Re: [EM] Can someone prove/disprove a relationship for me? [g is rational (II)(n) on X implies g is rational (I)(n) on X]

[David Catchpole]

Ignorance- the mark of the dilettante...

Ignore my idiotic statements about "concave" and "convex" sets- I
meant, respectively, union and intersection. In fact, ignore my last two
e-mails. I'm trying with maple this afternoon so I might not need to
pester you about this any longer.

Sorry...

On Mon, 22 Mar 1999, David Catchpole wrote:

> Where it says g*, substitute f
> 
> On Mon, 22 Mar 1999, David Catchpole wrote:
> 
> > General conditions:
> > 
> > Let X be some set. Let g be some function which returns some set over the
> > domain of all subsets of X, including X. Let f be some function such that
> > for any of the subsets Y of X, including X, f(Y) is equal to the
> > convex set over Y and g(Y) [that is, the set of all elements of Y which
> > are also elements of g(Y)]. Let n be some natural number.
> > _______________________________________________________________________________
> > Rational (I)(n):
> > 
> > g is rational (I)(n) on X if and only if;
> > 
> > -------------------------------------------------------------------------------
> > For any of the subsets Y of X, including X, for any of the subsets Z of X,
> > including X, for any of the subsets W of X, including W;
> > 
> > If;
> > 
> > f(Y) is any of the subsets of Z, including Z, AND W is any of the
> > subsets of Y, including Y, AND the cardinality of W [number of elements in
> > W] is greater than or equal to n, AND the cardinality of Z is greater
> > than or equal to n, AND;
> > 
> > f(V) is any of the subsets of W, including W, OR f(V) is any of the
> > subsets of f(Z), including f(Z), where V is equal to the concave set over
> > W and f(Z) [that is, the set of all elements of W or f(Z)];
> > 
> > Then it must be that;
> > 
> > f(V) is equal to f(Z), where V is equal to the concave set over W and
> > f(Z).
> > -------------------------------------------------------------------------------
> > _______________________________________________________________________________
> > 
> > _______________________________________________________________________________
> > Rational (II)(n):
> > 
> > g is rational (II)(n) on X if and only if;
> > 
> > -------------------------------------------------------------------------------
> > For any of the subsets Y of X, including X, for any of the subsets Z of X,
> > including X;
> > 
> > If;
> > 
> > f(Y) is any of the subsets of Z, including Z, AND the cardinality of Y 
> > [number of elements in Y] is greater than or equal to n, AND the 
> > cardinality of Z is greater than or equal to n, AND;
> > 
> > f(V) is any of the subsets of Y, including Y, OR f(V) is any of the
> > subsets of f(Z), including f(Z), where V is equal to the concave set over
> > Y and f(Z) [that is, the set of all elements of Y or f(Z)];
> > 
> > Then it must be that;
> > 
> > g*(V) is equal to g*(Z), where V is equal to the concave set over Y and
> > f(Z).
> > -------------------------------------------------------------------------------
> > _______________________________________________________________________________
> > 
> > _______________________________________________________________________________
> > THEOREM:
> > 
> > g is rational (II)(n) on X implies g is rational (I)(n) on X
> > _______________________________________________________________________________
> > 
> > I'm having BIG problems proving/disproving this theorem... Could someone
> > give me a hand?
> > 
> > Thanks,
> > 
> > David
> > 
> > 
> 
>