Bill,
> In this case, the containment set of 1 / [0, 0] is { }, the empty
set.
>
> Is this what you propose?
>
> If so, then I claim this is a containment failure because in the
projective real system 1/0 = oo, which is not the empty set.
> Therefore there is an inconsistency in the proposed theory.
You know that there is a few consistent mathematical theories of geometry.
Euclidean geometry:
At most one line can be drawn through any point not on a given line parallel to the given line in a planeLobacheviskian geometry:
At least two distinct lines can be drawn through any point not on a given line parallel to the given line in a planeBoth models are consistent.
Simolary, there may many consistent interval theories.
Set-based interval model: [1,1] / [0,0] = {} .
Containment set interval model: [1,1] / [0,0] = [-oo,+oo]
P1788 reserves a place for various interval models.
The requirement of interval model is that it extends Moore interval model of nonempty bounded intervals.
An implementation of interval models is called a flavour.
Only one of the flavours is described in full. It is set-based one.
Other flavours wait for their editors.
There is a reservation that was made especially keeping in mind possible containment-set flavour.
Initially floor([0,1/2])=[0,0] was considered a common evaluation.
Then we recollected that this may be not true in containment set interval model floor([0,1/2])=[-1,0].
So we changed definition of common evaluation so that floor{[0,1/2]) is no more common evaluation.
Each flavour can define floor{[0,1/2]) independently.
See example 5 at the end of subsection 7.2 .
General requirements on flavours are described in Section 7.
Are there some more issues there that may prevent developing containment-set flavour or other flavour you are developing in future ?
-Dima
----- Исходное сообщение -----
От: bill@xxxxxxxxxxx
Кому: vladik@xxxxxxxx, mhack@xxxxxxx, stds-1788@xxxxxxxxxxxxxxxxx
Отправленные: Воскресенье, 24 Ноябрь 2013 г 3:12:15 GMT +04:00 Абу-Даби, Маскат
Тема: Re: Motion 52: final "Expressions" text for vote
On 11/23/13 1:07 PM, Kreinovich, Vladik
wrote:
We have gone through these questions when we started the standard.
The range of a function over the interval is defined as the set of all possible values of the function when its arguments are in the range. If for some values within the range, the function is not defined, there are no values to add to the set.
Do you mean instead of the above:
The range of a function over the interval is defined as the set of all possible values of the function when its arguments are in the function's domain of definition.
That is:
f(X) = { f(x) | x \in X \cap D_f };
where X is an interval and D_f is f's domain of definition.
Therefore, the interval extension of any function returns the empty set { } for any argument outside the function's domain of definition.
In this case, the containment set of 1 / [0, 0] is { }, the empty
set.
Is this what you propose?
If so, then I claim this is a containment failure because in the
projective real system 1/0 = oo, which is not the empty set.
Therefore there is an inconsistency in the proposed theory.
Cheers,
Bill