Definitions for supremum
supre·mum

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1. supremumnoun

   (of a subset) the least element of the containing set that is greater or equal to all elements of the subset. The supremum may or may not be a member of the subset.

WikipediaRate this definition:0.0 / 0 votes

1. 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.

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1. Supremum

   In mathematics, the supremum of a subset S of a totally or partially ordered set T is the least element of T that is greater than or equal to all elements of S. Consequently, the supremum is also referred to as the least upper bound. If the supremum exists, it is unique meaning that there will be only one supremum. If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S. For instance, the negative real numbers do not have a greatest element, and their supremum is 0. The existence or non-existence of a supremum is often discussed in connection with subsets of real numbers, rational numbers, or any other well-known mathematical structure for which it is immediately clear what it means for an element to be "greater-than-or-equal-to" another element. The definition generalizes easily to the more abstract setting of order theory, where one considers arbitrary partially ordered sets. The concept of supremum coincides with the concept of least upper bound, but not with the concepts of minimal upper bound, maximal element, or greatest element. The supremum is in a precise sense dual to the concept of an infimum.

Anagrams for supremum »

1. supermum

How to pronounce supremum?

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Numerology

1. Chaldean Numerology

   The numerical value of supremum in Chaldean Numerology is: 2

2. Pythagorean Numerology

   The numerical value of supremum in Pythagorean Numerology is: 9

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References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Supremum
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Supremum
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=supremum