greatest lower bound - definition. What is greatest lower bound
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%ما هو (من)٪ 1 - تعريف

LEAST (RESP. GREATEST) OF MAJORING (RESP. MINORING) ELEMENTS OF A PARTIALLY ORDERED SET (NOT NECESSARILY EXISTING IN ALL SETS)
Supremum; Least upper bound; Greatest lower bound; Suprema; Infima; LUB; Lowest upper bound axiom; Smallest upper bound; Infimum; Infima and suprema; Supremum and infimum
  • supremum = least upper bound

greatest lower bound         
<theory> (glb, meet, infimum) The greatest lower bound of two elements, a and b is an element c such that c <= a and c <= b and if there is any other lower bound c' then c' <= c. The greatest lower bound of a set S is the greatest element b such that for all s in S, b <= s. The glb of mutually comparable elements is their minimum but in the presence of incomparable elements, if the glb exists, it will be some other element less than all of them. glb is the dual to least upper bound. (In LaTeX "<=" is written as sqsubseteq, the glb of two elements a and b is written as a sqcap b and the glb of set S as igsqcap S). (1995-02-03)
supremum         
infimum         
[?n'f??m?m]
¦ noun Mathematics the largest quantity that is less than or equal to each of a given set or subset of quantities. The opposite of supremum.
Origin
1940s: from L., lit. 'lowest part'.

ويكيبيديا

Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is a greatest element in P {\displaystyle P} that is less than or equal to each element of S , {\displaystyle S,} if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the least element in P {\displaystyle P} that is greater than or equal to each element of S , {\displaystyle S,} if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).

The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.

The concepts of infimum and supremum are close to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the set of positive real numbers R + {\displaystyle \mathbb {R} ^{+}} (not including 0 {\displaystyle 0} ) does not have a minimum, because any given element of R + {\displaystyle \mathbb {R} ^{+}} could simply be divided in half resulting in a smaller number that is still in R + . {\displaystyle \mathbb {R} ^{+}.} There is, however, exactly one infimum of the positive real numbers relative to the real numbers: 0 , {\displaystyle 0,} which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.