This inclusion is done until the set of models is closed, in the sense that it includes all least upper bounds of all sets of models it contains.
32.
The least upper bound of two numbers is the smallest number that is divided by both of them, i . e . the least common multiple of the numbers.
33.
This example also demonstrates that the existence of a least upper bound ( the number 0 in this case ) does not imply the existence of a greatest element either.
34.
The symbols " and " denote the least upper bound and greatest lower bound operations, respectively, in the complete Boolean algebra " B ".
35.
The construction of the real numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals.
36.
Note that while the ordering on " P " [ D ] is given by the subset relation, least upper bounds do not in general coincide with unions.
37.
A well-ordered set also has the least-upper-bound property, and the empty subset has also a least upper bound : the minimum of the whole set.
38.
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.
39.
The fact that such sequences exist means that the collection of all computable real numbers does not satisfy the least upper bound principle of real analysis, even when considering only computable sequences.
40.
One way is obvious : it is immediately clear from the definition that if a poset is complete then any set, including directed and finite subsets, has a least upper bound.
