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:
“The reality is that zero defects in products plus zero pollution plus zero risk on the job is equivalent to maximum growth of government plus zero economic growth plus runaway inflation.”
—Dixie Lee Ray (b. 1924)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)