partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a totally ordered set, like the real numbers, the concepts are the same. As an example, let be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from together with the set of integers and the set of positive real numbers ordered by subset inclusion as above. Then clearly both and are greater than all finite sets of natural numbers. Yet, neither is smaller than nor is the converse true: both sets are minimal upper bounds but none is a supremum. Least-upper-bound property The is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called . If an ordered set has the property that every nonempty subset of having an upper bound also has a least upper bound, then is said to have the least-upper-bound property. As noted above, the set of all real numbers has the least-upper-bound property. Similarly, the set of integers has the least-upper-bound property; if is a nonempty subset of and there is some number such that every element of is less than or equal to then there is a least upper bound for an integer that is an upper bound for and is less than or equal to every other upper bound for 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. An example of a set that the least-upper-bound property is the set of rational numbers. Let be the set of all rational numbers such that Then has an upper bound ( for example, or ) but no least upper bound in : If we suppose is the least upper bound, a contradiction is immediately deduced because between any two reals and (including and ) there exists some rational which itself would have to be the least upper bound (if )