max(), sum(), next()

[Ken Starks]

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 ?

 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.