Monday, January 29, 2007

inf (glb) and sup

from mathforum

inf means "infimum," or "greatest lower bound." This is
slightly different from minimum in that the greatest lower bound is
defined as:

x is the infimum of the set S [in symbols, x = inf (S)] iff:

a) x is less than or equal to all elements of S
b) there is no other number larger than x which is less than or equal
to all elements of S.

Basically, (a) means that x is a lower bound of S, and (b) means that
x is greater than all other lower bounds of S.

This differs from min (S) in that min (S) has to be a member of S.
Suppose that S = {2, 1.1, 1.01, 1.001, 1.0001, 1.00001, ...}. This
set has no smallest member, no minimum. However, it's trivial to show
that 1 is its infimum; clearly all elements are greater than or equal
to 1, and if we thought that something greater than 1 was a lower
bound, it'd be easy to show some member of S which is less than it.

So that's the difference between inf and min. It's worth noting that
every set has an inf (assuming minus infinity is okay), and that the
two concepts are the same for finite sets.

glb is another way of writing inf (sort for "greatest lower bound")

sup and lub, which are short for "supremum" and "least upper bound."

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