LUB Function
LUB Function

LUB Function

In mathematics, given a subset S of an ordered set T, the supremum of S is the least element of T that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). The supremum may, or may not, belong to the subset S. If S contains a greatest element, then that element is the supremum; and if not, then the supremum does not belong to the subset.

Suprema are often considered for subsets of real numbers, rational numbers, or any other well-known mathematical structures for which it is immediately clear what it means for an element to be "greater-or-equal" than another element. Nonetheless, the definition generalizes easily to the more abstract setting of order theory where one considers arbitrary partially ordered sets.

In any case, suprema must not be confused with minimal upper bounds, or with maximal or greatest elements. Some notes on these issues follow below.

In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S. An important property of the real numbers is its completeness: every nonempty set of real numbers that is bounded above has a supremum. If, in addition, we define sup(S) = −∞ when S is empty and sup(S) = +∞ when S is not bounded above, then every set of real numbers has a supremum (see extended real number line).

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