Re: 1/[0,2]=NaI
Kreinovich, Vladik wrote:
> I think there is a logical confusion here. An element x belonfgs to
> the set of all a for which P(a) is true if and only if P(x) holds.
>
> The word "all" in Sunaga's definition does not mean "for all".
>
> The statement (for all x \in [0,2]) y = 1/x is clearly false for all
> y, since it means that for all x, including x=1 and x=2, we must have
> y=1/x. In particular, this means that this must be true for x=1,
> which leads to y=1, and for x=2, which means y=1/2. Since no number
> can be equal to 1 and to 1/2, this mean no real number satisfies the
> above condition.
I agree. 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.
>
> An element z belongs to set Z of all { x/y | x in X, y in Y } is and
> onloy if there exist values x in X and y in Y for which z=x/y.
>
>
> ________________________________
>
> From: stds-1788@xxxxxxxx on behalf of Nate Hayes
> Sent: Tue 6/16/2009 2:05 PM
> To: STDS-1788@xxxxxxxxxxxxxxxxx
> Subject: Re: 1/[0,2]=NaI
>
>
>
> Ralph Baker Kearfott wrote:
>> Nate Hayes wrote:
>>> Arnold Neumaier wrote:
>>>> Nate Hayes wrote
>>>> (in: Re-submission of motion 5: multiple-format arithmetic):
>>>>
>>>>> Arnold Neumaier wrote:
>>>>>> Nate Hayes schrieb:
>>>>>>> Arnold Neumaier wrote:
>>>>>>>> Nate Hayes schrieb:
>> .
>> .
>> .
>>
>>> My position on this subject therefore allows you to compute
>>> 1/[0,2]=[1/2,Inf) in your range enclosure example, i.e., it allows
>>> to avoid NaI in this case because it implies
>>>
>>> 1/[0,2] = 1/[0,0] \union 1/(0,2]
>>> = {empty} \union [1/2,Inf)
>>> = [1/2,Inf)
>>>
>>
>> According to my understanding, THE ABOVE (taking the union of
>> limiting of point values) is the basic idea underlying csets, for
>> what it's worth.
>>> In the context of predicate logic, though, such an interpretation is
>>> not correct. For example, if I seek values of y such that
>>>
>>> (for all x \in [0,2]) y = 1/x
>>>
>>> is true, there is no value of y when x=0 to make the conditional
>>> equation true.
>>
>> But doesn't classical interval arithmetic (which I thought is the
>> basic thing we are standardizing) seek
>>
>> (the set of all y such that there exists an x \in [0,2] with
>> y=1/x) ??
>> That is different from what you have above.
>
> I probably should have added they are different but the "for all"
> predicate
> is the classical definition, e.g., Sunaga provides this defintion:
> "By the
> quotient of X to Y, provided that zero does not fall in Y, we mean the
> interval consisting of the set Z of all { x/y | x in X, y in Y }."
>
> Nate