Nate Hayes wrote:
Michel Hack wrote:
Nate: disjoint(A,B) = ( A \prec B ) or ( B \prec A )
Arnold: disjoint(A,B) = (A.sup<B.inf) or (B.sup<A.inf)
Both are probably incomplete with respect to the handling of Empty.
With Empty =[Nan,NaN], the formula
disjoint(A,B) = au<bl || bu<al || isEmpty(A) || isEmpty(B)
works with 754 arithmetic for all Motion 3 intervals A=[al,au] and
B=[bl,bu].
There is no need to evaluate this through two intermediate calls to
\prec, only to have them eliminated again by the compiler.
Yet if we define Disjoint logically as "no elements in common", then
Disjoint should be (vacuously) TRUE if at least one operand is Empty.
Why the if? There is universal agreement that this is the meaning
of being disjoint.
http://en.wikipedia.org/wiki/Disjoint_sets
This last observation is actually a strong motivation for defining
Disjoint as a required primitive.
I see it just the opposite:
this is strong motivation not to have disjoint as a special
primitive, or at least if it is that it should be defined in terms of
the preceeding relation so as to provide consistent results.
The standard definition -- that two sets are disjoint if they have no
common element -- is consistent, intuitive, and familiar to everybody.
It needs no additional definition in terms of a ''preceding'' relation
that until recently was unheard of by anybody working with intervals,
and that only works for the special case of intervals.
Vacuous truth is a hard concept even for experts and mathemeticans.
If you really believe that, you'd also promote the abolishion of the
empty set (a vacuous collection of objects), of the convention that
every set is a subset of itself (vacuous removal of objects from a
set), of the number zero (the number of items violating a vacuous truth),
or of the rule that one is justified to conclude anything from a false
statement.
In practice, these apparent oddities cause trouble only for beginners.
Any reasonable math student handles them habitually after the first
year, with an error rate not higher than for other slips.
Arnold Neumaier