Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is a greatest element in that is less than or equal to all elements of if such an element exists.^{ [1]} Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.^{ [1]}
The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to all elements of if such an element exists.^{ [1]} Consequently, the supremum is also referred to as the least upper bound (or LUB).^{ [1]}
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 similar 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 (not including ) does not have a minimum, because any given element of could simply be divided in half resulting in a smaller number that is still in There is, however, exactly one infimum of the positive real numbers: which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound.
Formal definition
A lower bound of a subset of a partially ordered set is an element of such that
- for all
A lower bound of is called an infimum (or greatest lower bound, or meet) of if
- for all lower bounds of in ( is larger than or equal to any other lower bound).
Similarly, an upper bound of a subset of a partially ordered set is an element of such that
- for all
An upper bound of is called a supremum (or least upper bound, or join) of if
- for all upper bounds of in ( is less than or equal to any other upper bound).
Existence and uniqueness
Infima and suprema do not necessarily exist. Existence of an infimum of a subset of can fail if has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique.
Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties.
If the supremum of a subset exists, it is unique. If contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to (or does not exist). Likewise, if the infimum exists, it is unique. If contains a least element, then that element is the infimum; otherwise, the infimum does not belong to (or does not exist).
Relation to maximum and minimum elements
The infimum of a subset of a partially ordered set assuming it exists, does not necessarily belong to If it does, it is a minimum or least element of Similarly, if the supremum of belongs to it is a maximum or greatest element of
For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element. For instance, for any negative real number there is another negative real number which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.
However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.
Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.
Minimal upper bounds
Finally, a 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 least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called Dedekind completeness.
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 lacks 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 ) or a member of greater than (if ). Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.
If in a partially ordered set every bounded subset has a supremum, this applies also, for any set in the function space containing all functions from to where if and only if for all For example, it applies for real functions, and, since these can be considered special cases of functions, for real -tuples and sequences of real numbers.
The least-upper-bound property is an indicator of the suprema.
Infima and suprema of real numbers
In analysis, infima and suprema of subsets of the real numbers are particularly important. For instance, the negative real numbers do not have a greatest element, and their supremum is (which is not a negative real number).^{ [1]} The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset of the real numbers has an infimum and a supremum. If is not bounded below, one often formally writes If is empty, one writes
Properties
The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets: Let the sets and scalar Define
- if and only if and otherwise ^{ [2]}
- ; the scalar product of a set is just the scalar multiplied by every element in the set.
- ; called the Minkowski sum, it is the arithmetic sum of two sets is the sum of all possible pairs of numbers, one from each set.
- ; the arithmetic product of two sets is all products of pairs of elements, one from each set.
- If then there exists a sequence in such that Similarly, there will exist a (possibly different) sequence in such that Consequently, if the limit is a real number and if is a continuous function, then is necessarily an adherent point of
In those cases where the infima and suprema of the sets and exist, the following identities hold:
- if and only is a lower bound and for every there is an with
- if and only is an upper bound and if for every there is an with
- If and then and
- If then and
- If then and
- and
- If and are nonempty sets of positive real numbers then and similarly for suprema ^{ [3]}
- If is non-empty and if then where this equation also holds when if the definition is used.^{ [note 1]} This equality may alternatively be written as Moreover, if and only if where if^{ [note 1]} then
Duality
If one denotes by the partially-ordered set with the opposite order relation; that is, for all declare:
For subsets of the real numbers, another kind of duality holds: where
Examples
Infima
- The infimum of the set of numbers is The number is a lower bound, but not the greatest lower bound, and hence not the infimum.
- More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the minimum of the set.
- If is a decreasing sequence with limit then
Suprema
- The supremum of the set of numbers is The number is an upper bound, but it is not the least upper bound, and hence is not the supremum.
In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.
One basic property of the supremum is
The supremum of a subset of where denotes " divides", is the lowest common multiple of the elements of
The supremum of a subset of where is the power set of some set, is the supremum with respect to (subset) of a subset of is the union of the elements of
See also
Wikimedia Commons has media related to Infimum and supremum. |
- Essential supremum and essential infimum
- Greatest element and least element – Element ≥ (or ≤) each other element
- Maximal and minimal elements – Element that is not ≤ (or ≥) any other element
- Limit superior and limit inferior (infimum limit)
- Upper and lower bounds – Majorant and minorant in mathematics
Notes
- ^ ^{a} ^{b} The definition is commonly used with the extended real numbers; in fact, with this definition the equality will also hold for any non-empty subset However, the notation is usually left undefined, which is why the equality is given only for when
References
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} Rudin, Walter (1976). ""Chapter 1 The Real and Complex Number Systems"". Principles of Mathematical Analysis (print) (3rd ed.). McGraw-Hill. p. 4. ISBN 0-07-054235-X.
- ^ Rockafellar & Wets 2009, pp. 1–2.
- ^ Zakon, Elias (2004). Mathematical Analysis I. Trillia Group. pp. 39–42.
- Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.
External links
- "Upper and lower bounds", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Breitenbach, Jerome R. & Weisstein, Eric W. "Infimum and supremum". MathWorld.