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]

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
> 
>