Supremum

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A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond).

In mathematics, the supremum (sup) 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 (lub or LUB). 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 (or does not exist). For instance, the negative real numbers do not have a greatest element, and their supremum is 0 (which is not a negative real number).

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 supremum is in a precise sense dual to the concept of an infimum.

Supremum of a set of real numbers

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 completeness: every nonempty subset of the set of real numbers that is bounded above has a supremum that is also a real number.

Examples

Simple

The "Supremum" or "Least Upper Bound" of the set of numbers 1, 2, 3 is 3. Although 4 is also an upper bound, it is not the "least upper bound" and hence not the "Supremum".

Advanced

sup { 1, 2, 3 } = 3
sup { x ∈ ℝ: 0 < x < 1 } = sup { x ∈ ℝ: 0 ≤ x ≤ 1 } = 1
sup { (-1)n - 1/n : n ∈ ℕ and 0 < n } = 1
sup { a + b: a ∈ A and b ∈ B } = sup(A) + sup(B)
sup { x ∈ ℚ: x2 < 2 } = √2

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

sup { f(t) + g(t): t ∈ A } ≤ sup { f(t): t ∈ A } + sup { g(t): t ∈ A }

for any functionals f and g.

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 under the affinely extended real number system.

sup ℤ = ∞
sup ∅ = -∞

If the supremum belongs to the set, then it is the greatest element in the set. The term maximal element is synonymous as long as one deals with real numbers or any other totally ordered set.

To show that a = sup(S), one has to show that a is an upper bound for S and that any other upper bound for S is greater than a. Equivalently, one could alternatively show that a is an upper bound for S and that any number less than a is not an upper bound for S.

Suprema within partially ordered sets

Least upper bounds are important concepts in order theory, where they are also called joins (especially in lattice theory). As in the special case treated above, a supremum of a given set is just the least element of the set of its upper bounds, provided that such an element exists.

Formally, we have: For subsets S of arbitrary partially ordered sets (P, ≤), a supremum or least upper bound of S is an element u in P such that

  1. x ≤ u for all x in S, and
  2. for any v in P such that x ≤ v for all x in S it holds that u ≤ v.

Thus the supremum does not exist if there is no upper bound, or if the set of upper bounds has two or more elements of which none is a least element of that set. It can easily be shown that, if S has a supremum, then the supremum is unique (as the least element of any partially ordered set, if it exists, is unique): if u1 and u2 are both suprema of S then it follows that u1 ≤ u2 and u2 ≤ u1, and since ≤ is antisymmetric, one finds that u1 = u2.

If the supremum exists it may or may not belong to S. If S contains a greatest element, then that element is the supremum; and if not, then the supremum does not belong to S.

The dual concept of supremum, the greatest lower bound, is called infimum and is also known as meet.

If the supremum of a set S exists, it can be denoted as sup(S) or, which is more common in order theory, by \bigvee S. Likewise, infima are denoted by inf(S) or \bigwedge S. In lattice theory it is common to use the infimum/meet and supremum/join as binary operators; in this case a ∨ b = sup {a, b} (and similarly ∧ for infima).

A complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).

In the sections below the difference between suprema, maximal elements, and minimal upper bounds is stressed. As a consequence of the possible absence of suprema, classes of partially ordered sets for which certain types of subsets are guaranteed to have least upper bound become especially interesting. This leads to the consideration of so-called completeness properties and to numerous definitions of special partially ordered sets.

Examples

The supremum of a subset S of (ℕ, |), where | denotes "divides", is the lowest common multiple of the elements of S.

The supremum of a subset S of (P, ⊆), where P is the power set of some set, is the supremum with respect to ⊆ (subset) of a subset S of P is the union of the elements of S.

Comparison with other order theoretical notions

Greatest elements

The distinction between the supremum of a set and the greatest element of a set may not be immediately obvious. The difference is that the greatest element must be a member of the set, whereas the supremum need not. 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 x, there is another negative real number x/2, 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, 0 is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

In general, this situation occurs for all subsets that do not contain a greatest element. In contrast, if a set does contain a greatest element, then it also has a supremum given by the greatest element.

Maximal elements

For an example where there are no greatest but still some maximal elements, consider the set of all subsets of the set of natural numbers (the powerset). We take the usual subset inclusion as an ordering, i.e. a set is greater than another set if it contains all elements of the other set. Now consider the set S of all sets that contain at most ten natural numbers. The set S has many maximal elements, i.e. elements for which there is no greater element. In fact, all sets with ten elements are maximal. However, the supremum of S is the (only and therefore least) set which contains all natural numbers. One can compute the least upper bound of a subset A of a powerset (i.e. A is a set of sets) by just taking the union of the elements of A.

Minimal upper bounds

Finally, a 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 mentioned above, the concepts are the same.

As an example, let S be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above. Then clearly both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum.

Least-upper-bound property

Main article: 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 S has the property that every nonempty subset of S having an upper bound also has a least upper bound, then S is said to have the least-upper-bound property. As noted above, the set ℝ of all real numbers has the least-upper-bound property. Similarly, the set ℤ of integers has the least-upper-bound property; if S is a nonempty subset of ℤ and there is some number n such that every element s of S is less than or equal to n, then there is a least upper bound u for S, an integer that is an upper bound for S and is less than or equal to every other upper bound for S. 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 ℚ, the set of rational numbers. Let S be the set of all rational numbers q such that q2 < 2. Then S has an upper bound (1000, for example, or 6) but no least upper bound in ℚ: If we suppose p ∈ ℚ is the least upper bound, a contradiction is immediately deduced because between any two reals x and y (including √2 and p) there exists some rational p', which itself would have to be the least upper bound (if p > √2) or a member of S greater than p (if p < √2). 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 P every bounded subset has a supremum, this applies also, for any set X, in the function space containing all functions from X to P, where f ≤ g if and only if f(x) ≤ g(x) for all x in X. For example, it applies for real functions, and, since these can be considered special cases of functions, for real n-tuples and sequences of real numbers.

See also

References

  • Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976.

External links

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