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On 07/28/2011 10:30 PM, Nate Hayes wrote:
Arnold Neumaier wrote:-- Not giving any specification for how intersection and union handle decorations means users will invent their own (probably wrong) rules, i.e., what is the standard? Examples were given that show how incorrect handling of decorations with intersection or union can lead to catastrophic failures, so this is not something that should be left up to users.I had given an example where Motion 27 gives erroneous results for decorated intersections: The expression f(x)= x/((x+1) intersect x^2) is undefined for any x in [1,3], but Definition 7 claims a safe answer for f([1,3]).Arnold, this is not any example of erroneous results. First of all: ( [1,3] + 1 ) intersect [1,3]^2 = [2,4] intersect [1,9] = [2,4] and [1,3] / [2,4] = [1/4,3/2]. The function is not undefined anywhere on [1,3].
Your argument proves nothing. For example, the expression is undefined for x=1, for x=2, and for x=3, although all these are in [1,3].
The fact that no exception occurred in your calculation shows that the calculation only says something about this particular interval, not (like everywhere else in range computations) about the possible values of elements from that interval.
But to be safe for the application of mathematical theorems, they must provide information about the functions evaluated, not just about the history of possible exceptions.
As I've noted several times before, giving another semantics for the intersection operation leads to catastophic failures in certain algorithms, like those depicted in the PDF attached to: http://grouper.ieee.org/groups/1788/email/msg03570.html
I consider this a misuse of the intersection operation, which happens to work in this particular case although there is no coherent semantics behind it. The mathematically correct way of treating your example (with identical results) was given by John Pryce some time ago.
It is far better to have no decorations at all for union and intersection than to have decorations based on a logically faulty basis.You have not demonstrated a faulty basis in providing the decorations; but IMHO give another example why they are necessary.
None of the issues raised in my critique was answered satisfactorily by your reply a few days ago. But I got tired of preaching deaf ears.
If your motion passes, I'll quit contributing to the standards discussion. Better no standard than one that doesn't work correctly. Arnold Neumaier