In article <[EMAIL PROTECTED]>, Ken Starks <[EMAIL PROTECTED]> wrote:
> David C. Ullrich wrote: > > > > > I don't see why you feel the two should act the same. > > At least in mathematics, the sum of the elements of > > the empty set _is_ 0, while the maximum element of the > > empty set is undefined. > > > > And both for good reason: > > > > (i) If A and B are disjoint sets we certainly want to > > have sum(A union B) = sum(A) + sum(B). This requires > > sum(empty set) = 0. > > > > (ii) If A is a subset of B then we should have > > max(A) <= max(B). This requires that max(empty set) > > be something that's smaller than everything else. > > So we give up on that. > > Do we give up? Really ? Erm, thanks. I was aware of all that below. If we're being technical what's below is talking about the sup and inf, which are not the same as max and min. More relevant to the present context, I didn't mention what's below because it doesn't seem likely that saying max([]) = -infinity and min([]) = +infinity is going to make the OP happy... > From wikipedia: http://en.wikipedia.org/wiki/Empty_set > (Uses wikipedia's LaTeX notation -- I hope those interested > are OK with that ) > > <quote> > Mathematics > > [edit] Extended real numbers > > Since the empty set has no members, when it is considered as a subset of > any ordered set, then any member of that set will be an upper bound and > lower bound for the empty set. For example, when considered as a subset > of the real numbers, with its usual ordering, represented by the real > number line, every real number is both an upper and lower bound for the > empty set.[3] When considered as a subset of the extended reals formed > by adding two "numbers" or "points" to the real numbers, namely negative > infinity, denoted -\infty\!\,, which is defined to be less than every > other extended real number, and positive infinity, denoted +\infty\!\,, > which is defined to be greater than every other extended real number, then: > > \sup\varnothing=\min(\{-\infty, +\infty \} \cup \mathbb{R})=-\infty, > > and > > \inf\varnothing=\max(\{-\infty, +\infty \} \cup \mathbb{R})=+\infty. > > That is, the least upper bound (sup or supremum) of the empty set is > negative infinity, while the greatest lower bound (inf or infimum) is > positive infinity. By analogy with the above, in the domain of the > extended reals, negative infinity is the identity element for the > maximum and supremum operators, while positive infinity is the identity > element for minimum and infimum. -- David C. Ullrich -- http://mail.python.org/mailman/listinfo/python-list
