Equivalent Definition
In addition to the definition above, there is an equivalent definition. A directed set is a set A with a preorder such that every finite subset of A has an upper bound. The above definition implies this one: the upper bound of the empty subset is any existing element of A, because A is nonempty; furthermore, as provable with an induction argument over the size of nonempty finite subsets, the upper bound of a finite subset may be obtained by finding upper bounds of pairs iteratively.
Read more about this topic: Directed Set
Famous quotes containing the words equivalent and/or definition:
“Distinctions drawn by the mind are not necessarily equivalent to distinctions in reality.”
—Thomas Aquinas (c. 1225–1274)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)