Re: 1/[0,2]=NaI
Nate Hayes wrote:
Arnold Neumaier wrote:
On the other hand, standard interval arithmetic is only concerned with
the problem
(for some x \in [0,2]) y = 1/x (**)
Again, (*) is a y-dependent predicate. It has the value true precisely
for all y in [1/2,inf]. Therefore, the natural interval answer to
this query is [1/2,inf], as computed by _my_ definition of interval
division.
Arnold, the "natural" interval answer [1.25,2] is also a valid solution to
your definition
I have never heard of a definition having a solution.
(**) is a y-dependent predicate, and as for other problems like solving
F(y)=0, which is also a y-dependent predicate, it is natural to ask
for an enclosure of the set of values satisfying the predicate.
since x \in [.5,.8] represents "some" x \in [0,2]. In fact,
any interval Y \subseteq [1/2,Inf) will satisfy your definition. What you
define is essentially the inner rounding.
No. _You_ are introducing the inner stuff as a smokescreen.
Like most users of interval analysis, _I_ am seeking outer
enclosures of the set of points satisfying (**).
[1/2,inf] is the best outer enclosure. It is obtained by evaluating
1/[0,2] = [1/2,inf] according to _my_ recipe.
Evaluating it by _your_ recipe gives the useless answer NaI.
Nate Hayes wrote:
>
> For example, the logical formula
> (for all x in [-4,-1]) [1,3] \union sqrt(x)
> is undefined (NaI) because it is not true for all x.
More precisely: This formula is meaningless, since a quantifier can
be applied only to logical propositions, not to sets.
Thus this is irrelevant for the interval standard.
Nate Hayes wrote:
>
> It is my mistake for sloppy notation. We can seek values of y to
> find a set Y so that
> (for all x \in [0,2])(there exists y in Y) y = 1/x. (*)
This is not a condition on y, since the y in (*) is a bound variable
without any meaning outside its scope. Thus seek these values is
meaningless.
You can, however, meaningfully seek a set Y so that (*) holds.
But since there is no such set, your search will be in vain.
In any case, it is not clear what this more precise version
of your statement should say in the original context.
It does not justify 1/[0,2]=NaI.
I think that, in the discussion on the list, you should adhere to
the usual precision of mathematical language, rather than leaving
to your readers the problem of making sense of your far too sloppy
formulations.
Arnold Neumaier