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Topic: Least upper bound

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	Supremum - Wikipedia, the free encyclopedia
		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.
		Least upper bounds are important concepts in order theory, where they are also called joins (especially in lattice theory).
		Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound.
	en.wikipedia.org /wiki/Least_upper_bound (1446 words)

	Supremum (Site not responding. Last check: 2007-10-21)
		In mathematics, the supremum of a given set is the least element which is greater than or equal to each element of the set.
		In any case, suprema must not be confused with minimal upper bound s, or with maximal or greatest element s.
		As in the special case treated above, a supremum of a given set is just the least element of the set of its upper bound s, provided that such an element exists.
	www.serebella.com /encyclopedia/article-Supremum.html (1375 words)

	Least upper bound axiom - Wikipedia, the free encyclopedia
		The least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis.
		The axiom says that if a nonempty subset of the real numbers has an upper bound, then it has a least such.
		The rational number line does not satisfy the LUB axiom and hence is not complete.
	en.wikipedia.org /wiki/Least_upper_bound_axiom (219 words)

	Upper bound - Wikipedia, the free encyclopedia
		In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S.
		On the other hand, a set may have many different upper and lower bounds, and hence one is usually interested in picking out specific elements from the sets of upper or lower bounds.
		This leads to the consideration of least upper bounds (or suprema) and greatest lower bounds (or infima).
	en.wikipedia.org /wiki/Upper_bound (242 words)

	lub (Site not responding. Last check: 2007-10-21)
		The least upper bound of a set S is the smallest b such that for all s in S, s <= b.
		The lub of mutually comparable elements is their maximum but in the presence of incomparable elements, if the lub exists, it will be some other element greater than all of them.
		Lub is the dual to greatest lower bound.
	www.linuxguruz.org /foldoc/foldoc.php?lub (169 words)

	Supremum (Site not responding. Last check: 2007-10-21)
		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 real number that is greater than or to every number in S.
		As in the special case treated a supremum of a given set is the least element of the set of upper bounds provided that such an element exists.
		As a consequence of the absence of suprema classes of partially ordered for which certain types of subsets are to have least upper bound become especially This leads to the consideration of so-called completeness properties and to numerous definitions of special ordered sets.
	www.freeglossary.com /Supremum (1585 words)

	least upper bound (Site not responding. Last check: 2007-10-21)
		If a set S has the property that every nonempty subset of S has an upper bound also has a least upper bound, then S is said to have the least upper bound property.
		An example of a set that lacks the least upper bound property is Q, the set of rational numbers.
		There is a corresponding 'greatest lower bound property'; an ordered set possesses the greatest lower bound property if and only if it also posseses the least upper bound property.
	www.yourencyclopedia.net /Least_upper_bound.html (830 words)

	What are the 'real numbers,' really? (Site not responding. Last check: 2007-10-21)
		Then S has many upper bounds -- for instance, 3 is an upper bound, and 2.24 is another upper bound, and 2.23607 is another upper bound.
		Any rational upper bound for S would have to be slightly higher than Ö5, and between that rational number and Ö5 we can always find still another rational number.
		Note that 1 is an upper bound for S, and 1/2 is another upper bound for S, and 1/3 is another upper bound for S, and so on.
	www.cartage.org.lb /en/themes/Sciences/Mathematics/calculus/realnumbers/Least/least.htm (473 words)

	Definition of Upper Bound and Least Upper Bound (Supremum) (Site not responding. Last check: 2007-10-21)
		S. The set S is said to be "bounded above" by C. A function, f, is said to have a upper bound C if f(x) ≤ C for all x in its domain.
		The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if ε is any positive quantity, however small, there is a member that exceeds M - ε.
		The least upper bound of a function, f, is defined as a quantity M such that f(x) ≤ M for all x in its domain, but if ε is any positive quantity, however small, there is an x in the domain such that f(x) exceeds M - ε.
	www.mcraeclan.com /MathHelp/CalculusLimitUpperBound.htm (457 words)

	Encyclopedia: Domain theory (Site not responding. Last check: 2007-10-21)
		In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually.
		In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound.
		A poset D with a least element is a dcpo iff every monotone function f on D has a fixed point.
	www.nationmaster.com /encyclopedia/Domain-theory (4000 words)

	sciforums.com - Upper Bound
		I am required to prove that 3 is an upper bound for S and justify whether I think it is a least upper bound.
		An upper bound is defined as being a number in N such that for all s in S, k >= s.
		That would mean that no number less than 3 could be an upper bound since we could take "epsilon"= 3 minus that number and show that there must be a member of the sequence closer to 3 than that: i.e.
	www.sciforums.com /showthread.php?t=33574 (1328 words)

	GREATEST LOWER BOUND - Definition (Site not responding. Last check: 2007-10-21)
		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.
	www.hyperdictionary.com /computing/greatest+lower+bound (131 words)

	Upper bound (Site not responding. Last check: 2007-10-21)
		In mathematics especially in order theory an upper bound of a subset S of some partially ordered set is an element which is greater or equal to every element of S.
		On the other hand a set have many different upper and lower bounds hence one is usually interested in picking specific elements from the sets of upper lower bounds.
		I was very impressed with the way this publication was set up.Maps of the Ontario Counties and Districts, to accompany the statis of the Loyalist and generations proceeding.I would recommend this to enyone looking for the generations that belong to the Lo...
	www.freeglossary.com /Upper_bound (372 words)

	Least primitive root of prime numbers
		The least primitive root computation is done in assembly language, to take advantage of some floating point capabilities of the processor (this alone resulted in a very significant speedup of our program).
		(The region between these two bounds is clearly marked in the appropriate figure.) It is interesting to observe that the upper bound appears to be close to the square of the lower bound.
		Least prime base required to prove the primality of a number
	www.ieeta.pt /~tos/p-roots.html (1612 words)

	Least Upper bounds (Site not responding. Last check: 2007-10-21)
		On the semantics of lower and upper bounds...
		Least Upper Bounds for the Size of OBDDs Using Symmetry Properties...
		Upper Bounds for the Derivative of Exponential Sums...
	www.scienceoxygen.com /math/354.html (265 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The element $a$ of $X$ is called the {\bf supremum} or {\bf least upper bound} of $S$ if \begin{enumerate} \item[(i)] $a$ is itself an upper bound of $S$, and \item[(ii)] no element $b$ of $X$ which is less than $a$ can be an upper bound for $S$.
		In practice, when one uses the least upper bound property, one always has to remember to check that the set being considered is nonempty.
		In some theorems, one wants to use the least upper bound property to show that a set has a supremum, and it turns out to be quite hard to check that the set is non-empty.
	www.math.unl.edu /~jorr/webnotes/src/classes-1997/class05.wfy (586 words)

	bounded - definition by dict.die.net (Site not responding. Last check: 2007-10-21)
		bounded adj : having the limits or boundaries established; "a delimited frontier through the disputed region" [syn: delimited]
		To limit; to terminate; to fix the furthest point of extension of; -- said of natural or of moral objects; to lie along, or form, a boundary of; to inclose; to circumscribe; to restrain; to confine.
		bounded In domain theory, a subset S of a cpo X is bounded if there exists x in X such that for all s in S, s
	dict.die.net /bounded (99 words)

	Analysis WebNotes: Chapter 02, Class 06
		It is now time to show that the rationals do not have the least upper bound property, and also that there are real numbers which are not rational.
		What about lower bounds and the greatest lower bound (which is the corresponding notion to the least upper bound).
		Since a is an upper bound for the set in the rationals, it is also an upper bound for the set in the reals, and so the square root of 2 is less than or equal to it.
	www.math.unl.edu /~webnotes/classes/class06/class06.htm (816 words)

	Lattice Model of Information Flow (Site not responding. Last check: 2007-10-21)
		There is a highest class H, which is the least upper bound of all classes, and a least class L, which is the greatest lower bound of all classes.
		The least upper and greatest lower bound properties greatly simplify the amount of information needed to track the origins and destinations of flows.
		A universally bounded lattice is a structure consisting of a finite partially ordered set together with least upper and greatest lower bound operators on the set.
	courses.cs.vt.edu /~cs5204/fall99/distributedSys/lattice.html (1457 words)

	Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to ...
		Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model, Hiroshi Kurata, Takeaki Kariya
		Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model
		In a general normal regression model, this paper first derives the least upper bound (LUB) for the covariance matrix of a generalized least squares estimator (GLSE) relative to the covariance matrix of the Gauss-Markov estimator.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aos/1032298283 (320 words)

	# 177 - Meaning of"least upper bound"and"greatest lower bound"
		The erroneous use of the terms "least upper bound" and "greatest lower bound" conveys the notion that disjoint pairs of categories are omitted - or excluded - from the policy.
		There exist pairwise upper bounds in the set of security attributes, such that, given any two valid security attributes, there is a valid security attribute that is greater than or equal to the two valid security attributes; and
		There exist pairwise lower bounds in the set of security attributes, such that, given any two valid security attributes, there is a valid security attribute for which both of the two original valid security attributes are greater than the security attribute.
	niap.nist.gov /cc-scheme/interpretations/draft177.html (434 words)

	Real Numbers
		An element b is called an upper bound for the set X if every element in X is less than or equal to b.
		An element b in A is called a least upper bound (or supremum) for X if b is an upper bound for X and there is no other upper bound b' for X that is less than b.
		Again, as for the Least Upper Bound property, it is more important to understand what these properties mean than to follow the proof exactly.
	pirate.shu.edu /projects/reals/infinity/reals.html (813 words)

	Axioms for the Real numbers
		An upper bound of a non-empty subset A of R is an element b
		A lower bound of a non-empty subset A of R is an element d
		Thus b - 1/n is an upper bound, contradicting the assumption that b was the least upper bound.
	www.gap-system.org /%7Ejohn/analysis/Lectures/L5.html (1036 words)

	Mathwords: Least Upper Bound of a Set
		The smallest of all upper bounds of a set of numbers.
		For example, the least upper bound of the interval (5, 7) is 7.
		The least upper bound of [5, 7] is also 7.
	www.mathwords.com /l/least_upper_bound.htm (43 words)

	[No title]
		Back in the #ref(class05)last section we asserted that the rational numbers do not have the least upper bound property, and the reals do have this property.
		The element $a$ of $X$ is called the {\bf infimum} or {\bf greatest lower bound} of $S$ if \begin{enumerate} \item[(i)] $a$ is itself a lower bound of $S$, and \item[(ii)] no element $b$ of $X$ which is greater than $a$ can be a lower bound for $S$.
		The idea will be to show that $B$ has a supremum (using the least upper bound property of $X$) and that this supremum does the job of the infimum of $S$.
	www.math.unl.edu /~webnotes/src/classes-1997/class06.wfy (1796 words)

	Encyclopedia: Dedekind cut (Site not responding. Last check: 2007-10-21)
		If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S.
		Regard one Dedekind cut { A, B } as less than another Dedekind cut { C, D } if A is a proper subset of C, or, equivalently D is a proper subset of B.
		In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound.
	www.nationmaster.com /encyclopedia/Dedekind-cut (1079 words)

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