Re: Empty interval representations & Motion 13...
On 2010-04-21 19:46:24 -0700, Dan Zuras Intervals wrote:
> Folks,
>
> I have been thinking about something for some time now
> & Ulrich's recent revision of Motion 13 has brought it
> up again.
>
> It is the issue of the representation of the empty
> interval.
[...]
Anyway Motion 13-2 doesn't specify some cases when a and/or b is the
empty interval, e.g. for <=. This should be done before talking about
the representation. I don't understand "The greatest lower bound and
the least upper bound of an interval with the empty set are both the
empty set." How can this is possible?
> Let's let any = [b1,b2] be some otherwise non-empty
> interval. I believe we have that
>
> empty equals any = False
> empty subset any = True
> empty lessEqual any = False (1)
> empty precedeTouch any = True (2)
> empty interior any = True
> empty strictLess any = False
> empty preceed any = True (3)
>
> as well as
>
> any equals empty = False
> any subset empty = False
> any lessEqual empty = False (1)
> any precedeTouch empty = True (2)
> any interior empty = False
> any strictLess empty = False
> any preceed empty = True (3)
>
> and, for completeness sake
>
> empty equals empty = True
> empty subset empty = True
> empty lessEqual empty = True (4)
> empty precedeTouch empty = True (2)
> empty interior empty = False
> empty strictLess empty = False
> empty preceed empty = True (4)
Replacing "any" by "non-empty" would be more clear.
One problem is that "a precedes b" doesn't imply "a lessEqual b".
There may be others...
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)