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All 3 problems point to the difficulty of "identity" when dealing with intervals. Suppose we have two numbers A and B which are contained in intervals:
A = [2, 8]
B = [2, 8]
A could be any number, such as 3.782, and B could be 5.42, but we just don't know.
Now, A divided by itself must be 1.0 (assuming A isn't 0), but of A/B (the same applies to subtraction) we can only say that it's somewhere in the interval
Unfortunately, our interval package doesn't say anything about identity, so if we calculated A/A, we would also get
So, any time we do algebraic manipulation of an equation involving intervals, we need to be careful any time we introduce the same interval (e.g. through fraction reduction), since our interval package re-introduces the uncertainty, even if it shouldn't.
2.14. Lem just demonstrates the above.
2.15. Eva is right, since the error isn't reintroduced into the result in par2 as it is in par1.
2.16. A fiendish question. They say it's "very difficult" as if it's doable. I'm not falling for that. Essentially, I believe we'd have to introduce some concept of "identity", and then have the program be clever enough to reduce equations. Also, when supplying arguments to any equation, we'd need to indicate identity somehow, since [2, 8] isn't necessarily the same as [2, 8] ... unless it is. Capiche?
A better explanation and some pointers to the interesting world of interval arithmetic