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# Topic: Bounded set

 Boundedness - Wikipedia, the free encyclopedia The term bounded appears in different parts of mathematics where a notion of "size" can be given. A linear transformation is bounded if the image of the unit ball is a bounded set. A partially ordered set is bounded if it is has both a greatest element and a least element. en.wikipedia.org /wiki/Bounded   (253 words)

 Bounded set - Wikipedia, the free encyclopedia Therefore, a set is bounded if it is contained in a finite interval. M is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. en.wikipedia.org /wiki/Bounded_set   (286 words)

 Metric space - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-12) In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded. a bounded set is referred to as "a finite interval" or "finite region". www.bucyrus.us /project/wikipedia/index.php/Metric_space   (1472 words)

 Sets are sets and the objects in the set denoted by is a subset of the set denoted by Sets are mutually-disjoint if and only if there is no object that is a member of all of the sets. logic.stanford.edu /kif/Hypertext/node23.html   (511 words)

 The MSc in Computer Science A hereditarily finite set is a finite set whose elements are... the set x = {x} = {{{...}}} corresponds to the graph with one node and one "self-looping" arc. It does require a combination of understanding the underlying concepts with implementation skills, so it should be seen as of medium to high level of difficulty, depending on how ambitiously it is pursued. web.comlab.ox.ac.uk /oucl/courses/grad03-04/cs/proj3.html   (407 words)

 Existence of an optimum   (Site not responding. Last check: 2007-10-12) The difficulties in the first two cases are that the set S is unbounded; the difficulty in the third case is that the interval S is open (does not contain its endpoints); and the difficulty in the last case is that the function f is discontinuous. 1} is not bounded, because for any number k the point (2k, 0) is in the set, and the distance of this point from (0, 0) is 2k, which exceeds k. A continuous function on a compact set attains both a maximum and a minimum on the set. www.chass.utoronto.ca /~osborne/MathTutorial/OPN.HTM   (557 words)

 On Bounded Set Theory - Yu (ResearchIndex)   (Site not responding. Last check: 2007-10-12) 1.5: A Bounded Set Theory with Anti-Foundation Axiom and Inductive.. 1 A bounded set theory with anti-foundation axiom and inductiv.. A Bounded Set Theory with Anti-Foundation Axiom and Inductive.. citeseer.ist.psu.edu /103440.html   (872 words)

 shift happens! - View topic - Is Jesus Fuzzy?!..or Bounded?.. Or Centered?   (Site not responding. Last check: 2007-10-12) From this perspective, a set of people who claim to have some connection to Christ can be shown to be part of the set by ascertaining whether their beliefs and behaviours are within certain set boundaries. Christendom as a culture was a bounded set, maintained by strict control structures to ensure everyone within its boundaries believed and behaved correctly and by political or military action to defend it against those who did not belong. The emerging church is incapable of becoming a bounded set, as there is simply (A) too broad of a spectrum in the conversation, and (B) there is no "head office" (including Emergent Village) that is trying to monolithically enforce a set pattern, doctrinal statement, or missional approach. www.3dff.com /php/viewtopic.php?t=386   (11823 words)

 A sequence of decreasing non-empty compact sets has a non-empty compact intersection   (Site not responding. Last check: 2007-10-12) is bounded since it is a subset of the bounded set and is closed, since it is the intersection of a family of closed sets. is infinite, it is an infinite bounded set, so has at least one cluster point. www.math.pitt.edu /~sparling/23014/23014notes6/node24.html   (97 words)

 Boundedness   (Site not responding. Last check: 2007-10-12) The subset axiom assures that the set of all of subsets of a bounded set is also a bounded set. Since every finite set is bounded, this allows us to conclude, as a special case, that the union of any finite number of bounded sets is a bounded set. There is a bounded set containing a set, a set that properly contains that set, a third set that properly contains the second set, and so forth. logic.stanford.edu /kif/Hypertext/node24.html   (285 words)

 Math Forum - Ask Dr. Math Date: 06/24/2003 at 14:57:11 From: Doctor Mike Subject: Re: Proof about a bounded set Hi, The infimum is the LARGEST possible lower bound, and the supremum is the SMALLEST possible upper bound, so it is good to use those words only when you mean exactly those specific concepts. Assume for notation purposes that the positive "M" is a bound for S. Note, it may NOT be the SMALLEST bound for S. What we are required to prove to show that (b) is true is that S has an upper bound and has a lower bound. You correctly observed that M must be an upper bound, and -M must be a lower bound. mathforum.org /library/drmath/view/63679.html   (466 words)

 PlanetMath: bounded set (in a topological vector space) "bounded set (in a topological vector space)" is owned by mathcam. Cross-references: bounded, compact set, theorem, scalar, zero vector, neighborhood, topological vector space, subset This is version 4 of bounded set (in a topological vector space), born on 2003-07-07, modified 2005-04-05. planetmath.org /encyclopedia/BoundedSetInATopologicalVectorSpace.html   (101 words)

 Metric space Article, Metricspace Information   (Site not responding. Last check: 2007-10-12) In mathematics, a metric space is a set (or "space") where a distance between points is defined. Since the set of the centres of these balls is finite, it has finite diameter, from which it follows(using the triangle inequality) that every totally bounded space is bounded. A simple way to construct a function separating a point from a closed set (as required for a completely regular space) is to consider the distance between the pointand the set. www.anoca.org /spaces/set/metric_space.html   (1572 words)

 Definition of Upper Bound and Least Upper Bound (Supremum) 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)

 422Notes If S is a bounded set, show that there is an increasing subsequence of S that converges to sup(S) and a decreasing subsequence of S that converges to inf(S). This says that the set of distinct points of the sequence is finite and thus no neighborhoods of any point can contain infinitely many distinct points of the sequence. Further, if a set is bounded and has exactly one cluster point but is not closed, then it does not contain its cluster point. www.puc.edu /Faculty/George_Hilton/id94.htm   (1056 words)

 Homework 1   (Site not responding. Last check: 2007-10-12) Every non-empty bounded set has an infimum and a supremum. Every non-empty set, which is bounded from below, has an infimum. Every non-empty set, which is bounded from above, has a supremum. www.math.utep.edu /Faculty/helmut/oldclass/974_3341/hw1/hw1.html   (116 words)

 The weak-* topology in subsystems of Z2 In the comprehension scheme it is assumed that the set variable W does not occur freely in A bounded linear functional on X is a bounded linear operator Thus the existence of a sequence of finite sets www.math.psu.edu /simpson/papers/convex-l/node4.html   (1224 words)

 Mass Problems and Randomness The set of all recursively enumerable Turing degrees is denoted sets, there is reason to view weak reducibility as the mass problem analog of Turing reducibility, while strong reducibility is the mass problem analog of truth table reducibility. sets of positive measure, there is no largest or even maximal degree. www.math.psu.edu /simpson/papers/massrand/massrand.html   (2011 words)

 Science Math Logic and Foundations Set Theory   (Site not responding. Last check: 2007-10-12) Bounded Set Theory - A weak version of ordinary set theory using bounded quantification. Set Theory - Survey from the Stanford Encyclopedia of Philosophy by Thomas Jech. Sets and Their Sizes - An alternative to Cantor's theory of cardinality. www.iper1.com /iper1-odp/scat/id/Science/Math/Logic_and_Foundations/Set_Theory   (502 words)

 About "Bounded Set Theory"   (Site not responding. Last check: 2007-10-12) Bounded Set Theory (BST) is a weak version of the ordinary set theory. Its main feature is paying main attention to using bounded quantification (as in the ordinary everyday mathematical practice) and other analogous bounded constructs. The main universe of sets for BST consists of hereditarily-finite sets and is called HF. mathforum.org /library/view/17320.html   (107 words)

 Remarks on Proving The Fundamental Theorem of Algebra We first prove the Bounded Value Theorem – the range of a continuous function on a compact set is bounded. Now proceed by successive bisection: bisect the original compact set (here is where we use the boundedness); on at least one piece, the function is unbounded. Since the function is bounded, there is a least upper bound, say M, for the range of the function. www.cut-the-knot.org /fta/fta_note.shtml   (311 words)

 ► » bounding an infinate bounded convex set by polyhedra   (Site not responding. Last check: 2007-10-12) bounding an infinate bounded convex set by polyhedra Subject: bounding an infinate bounded convex set by polyhedra that your set S (and therefore its convex hull C) is contained in the www.science-chat.org /bounding-an-infinate-bounded-convex-set-by-polyhedra-6884822.html   (199 words)

 [No title]   (Site not responding. Last check: 2007-10-12) Note: a set will be compact if it is closed and bounded. Since the indicated set is bounded and contains all its limit points, it is compact. Since the countable union of finite sets is countable, \$S\$ itself is countable. www.mcs.drexel.edu /~rboyer/courses/analy.1/win00/hw.1.soln   (621 words)

 [No title]   (Site not responding. Last check: 2007-10-12) To define a regular set, recall that the interior of a set includes all points in the set such that there is a small ball around them which is contained in the set. The closure of a set includes all points which are the limit of sequences of points in the set. Finally, a set is semi-algebraic if it can be expressed as a finite union of a finite number of intersections of algebraic half-spaces. www.cs.huji.ac.il /~danix/modeling/lecture2.html   (1170 words)

 complete.htm Analysis Fact Sheet A, then f(z) is bounded on A. If f(z) is continuous on a closed and bounded set A, then f(z) is uniformly continuous on A. Uniformly continuity on A means that there a function A peculiar viewpoint of closed would be: A set A of complex numbers is closed if it has the Bolzano-Weierstrass property when considered as an entity in itself. A closed and bounded set is a compact set www.math.uic.edu /~lewis/hon201/complete.htm   (505 words)

 Analysis WebNotes: Chapter 06, Class 31 Sequentially compact sets are important because continuous functions defined on sequentially compact sets have some very useful properties, which they do not have in general when defined on non-compact sets. In general the continuous image of closed sets isn't closed, but this example shows that quite often continuous functions on Euclidean space (in particular, on the real line) will map closed set to closed sets. In Corollary 6.8 we saw that a continuous function defined on a sequentially compact set is always bounded. www.math.unl.edu /~webnotes/classes/class31/class31.htm   (669 words)

 ISO/IEC 10744 - 6.5 HyTime Bounded object set Support for the management of HyTime bounded object sets is managed through the HyTime bos and bosspcat attribute forms of the HyDoc element form, the bosdatt data attribute form, and the bosspec element form. ) attribute sets the default priority for itself and for all of the entities in the entity subtree of which it is the root. NOTE 101 For example, it may be necessary to comply with activity policies appearing in objects that are external to the hub document; such objects should be specified as having foreground priority. www.y12.doe.gov /sgml/wg8/docs/n1920/html/clause-6.5.html   (2140 words)

 Compactness The property of being a bounded set in a metric space is not preserved by homeomorphism. A compact subset of R with its usual metric is closed and bounded. The closed bounded interval is compact and hence its image is compact and hence is also a closed bounded subset which is in fact an interval also, by connectedness. www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L21.html   (704 words)

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