Standard definition (where X = [a, b] and Y = [c, d]):
X < Y <=> b < c X > Y <=> a > d
There are two possible comparisons for intervals. These are certainly and possibly relations. If you want to make sure that X > Y then you should use the certainly relation, which means that a > d.
If you want you can use possibly greater than comparison by checking b > c.
Less than relations are similar to greater than. I hope this helps you.
Another way of saying this might be: it depends on whether you interpret an interval to mean "there is a point that lies between the bounds" and the relation between two intervals to be a relation between the two points ("possibly" relations) or whether you mean by an interval "the set of all points between the bounds" and the relation between two intervals to be a relation between two sets ("certainly" relations).
Cheers - André Vellino
The definition of an interval must be invariant.
Formal definitions for all certainly, possibly, *and* set relations are contained in a specification for support of intervals in Fortran. Earlier drafts of this specification have been posted to www.mscs.mu.edu/~globsol/walster-papers.html
A few implementation issues are in the process of being clarified in the final draft, which will also be posted to the above URL. With respect to the current question about the interpretation of ordering relations between intervals, the draft posted above will suffice.
Regards to all,
All cases of two intervals comparing were investigated in my paper: Alexander G. Yakovlev, Classification approach to programming of localizational (interval) computations, Interval Computations, 1(3), 1992, pp. 61-84. If don't have this issue, I can send you its TeX version.
If you have a question related to validated computing, interval analysis, or related matters, I recommend