of subsets of S, with union is S, which are non-empty, and pairwise disjoint. This definition differs from a partition of a set, in that the order of the Ai matters.
For example, one ordered partition of { 1, 2, 3, 4, 5 } is
{1, 2} {3, 4} {5}
which is equivalent to
{1, 2} {4, 3} {5}
but distinct from
{3, 4} {1, 2} {5}.
The number of ordered partitions Tn of { 1, 2, ..., n } can be found recursively by the formula:
In mathematics, a Dedekind cut in a totally orderedset S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is closed upwards.
In this way, the set of all Dedekind cuts is itself a linearly orderedset, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound.
More generally, in a partially orderedset S, the set of all nonempty downwardly closed subsets (also called order ideals) is a set partially ordered by inclusion, and in the same way we embed S within a larger partially orderedset that, generally unlike the original set S, does have the least-upper-bound property.
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