Re: [IEEE P1788 er subgroup]: My first cut at the Level 1 list...
> Date: Tue, 31 Mar 2009 15:28:53 +0200
> From: Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
> CC: stds-1788-er <stds-1788-er@xxxxxxxxxxxxxxxxxxxxx>,
> stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: [IEEE P1788 er subgroup]: My first cut at the Level 1 list...
>
> . . .
>
> >
> > One of your countrymen once said, "A theory should be as simple
> > as possible, and no simpler."
> >
> > One must admit that there is a certain lower bound on simplicity
> > when it comes to intervals. :-)
> >
> > Thanks,
> >
> >
> > Dan
> >
> >
> Dan:
>
> I really enjoy your answer. It reads so philosophical. Let me try to
> give an answer in terms of mathematics.
>
> For intervals A and B in IR (the set of closed and bounded real
> intervals) division is defined by
> I. A/B := {a/b | a in A and b in B} for 0 notin B.
> The quotient a/b is defined as the inverse operation of multiplication,
> i.e., as the solution of the equation b x = a. Thus the above definition
> can be written in the form
> II. A/B := {x | b x = a for all a in A and b in B}.
> For 0 notin B both definitions are equivalent. While in R division by
> zero is not defined, representation of A/B in the second form allows
> definition of the operation and also interpretation of the result for 0
> in B.
>
> Since every x in R fulfills the equation 0 x = 0 we obtatin A/B = (-oo,
> +oo) whenever 0 in A and 0 in B. So 0/0 = (-oo, +oo).
>
> If 0 notin A and B = [0, 0] there is no real number which fulfills the
> second form of the definition of the quotiont. So in this case A/B =
> empty set.
>
> So the answer in case of your samples is clear:
> 1/[0, 0] = emptyset
> and for X = [1, 1], 1/(X - 1)^2 = emptyset as well as 1/(X -1) =
> emptyset.
>
> Interval arithmetic deals with sets of real numbers. In addtion to the
> arithmetic operations, for sets we still have the operations
> intersection and union. So large intervals can become smaller within a
> computation by intersection. In interval arithmetic the emptyset playes
> a role comparable to NaN in floating-point arithmetic. However, this
> last sentence is also more of a philosophical nature.
>
> For more details see my position paper "Complete Interval Arithmetic and
> Its Implementation on the Computer".
>
> Best whishes
> Ulrich
>
Ulrich,
I am happy with this as an answer. It is in keeping with what
I have learned about c-sets. There may be some difficulty
when it comes to non-standard intervals but we can work that
out when we come to it.
Thus, as I believe Michel pointed out, while it looks like
having R* as our basis & the two elements [+oo,+oo] & [-oo,-oo]
amount to the same thing they can be separated without any (or
much) difficulty.
And it is in keeping with my desire to ruthlessly exclude NaNs
from our consideration as elements of intervals. (That way
lies madness. :-)
For what is a philosopher but a mathematician who can't count.
Enjoy,
Dan