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# Topic: Union set theory

###### In the News (Wed 22 May 13)

 Union (set theory): Definition and Links by Encyclopedian.com - All about Union (set theory)   (Site not responding. Last check: 2007-11-06) In set theory and other branches of mathematics, the union of some sets is the set that contains everything that belongs to any of the sets, but nothing else. The empty set is an identity element for the operation of union. That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in formal set theory. www.encyclopedian.com /un/Union-(sets).html   (687 words)

 Axiomatic set theory   (Site not responding. Last check: 2007-11-06) Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundations of mathematicsfoundational theory/ in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. It is often asserted that axiomatic set theory is thus an adequate foundation for current mathematical practice, in the sense that ''in principle'' all proofs produced by the mathematical community could be written formally in set theory terms. www.infothis.com /find/Axiomatic_set_theory   (2596 words)

 Union (set Theory) Encyclopedia Article, Definition, History, Biography   (Site not responding. Last check: 2007-11-06) In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. For example, the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in axiomatic set theory. www.karr.net /search/encyclopedia/Union_%28set_theory%29   (843 words)

 PlanetMath: concepts in set theory The aim of this entry is to present a list of the key objects and concepts used in set theory. The reader should take care that if the objects under discussion are not just sets (say, groups or schemes) the operations may not be simple set operations, but rather their analogue in the relevant category. This is version 44 of concepts in set theory, born on 2004-02-29, modified 2005-04-14. planetmath.org /encyclopedia/NotationInSetTheory.html   (357 words)

 Set Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as elementary school. The set of all natural numbers is a proper subset of all integers. www.askfactmaster.com /Set   (851 words)

 set - a Whatis.com definition - see also: set theory   (Site not responding. Last check: 2007-11-06) Sets are usually symbolized by uppercase, italicized, boldface letters such as A, B, S, or Z. Each object or number in a set is called a member or element of the set. Set theory is fundamental to all of mathematics. However, set theory is closely connected with symbolic logic, and these fields are becoming increasingly relevant in software engineering, especially in the fields of artificial intelligence and communications security. searchsecurity.techtarget.com /sDefinition/0,,sid14_gci333100,00.html   (425 words)

 Set Theory   (Site not responding. Last check: 2007-11-06) For instance, the set G consisting of the 10 most enjoyable things is not a well-defined set, because it is difficult to pinpoint what exactly is in this set, and the contents of this set would certainly also differ from person to person. On the other hand, the set D of all the dogs in my yard, is certainly well-defined because I can determine the elements of the set with a relatively high degree of accuracy. The intersection is the set theoretic sibling to 'and' in logic, and the symbol we use reflects this. www.math.ou.edu /~rkirkpat/settheor.htm   (921 words)

 Set theory All sets are therefore subsets of the universal set. The set A is therefore a subset of the universal set. AuB means the union of sets A and B and contains all of the elements of both A and B. www.projectalevel.co.uk /maths/settheory.htm   (216 words)

 Encyclopedia article on Set [EncycloZine]   (Site not responding. Last check: 2007-11-06) A set may be defined by specifying in words the property which characterizes it, and enclosing this description in curly braces. Such a set is called the empty set (or the null set) and is denoted by the symbol Template:Unicode. A set can also have an infinite number of members; for example, the set of natural numbers is infinite. encyclozine.com /Set   (1243 words)

 80.07.04: Logic and Set Theory A set is a well defined collection of “objects.” The term “well defined” means that the set is described in such a way that we can determine whether or not any given object belongs to that set. Note: The definition for union says that in order for an element to be part of the solution set for AU B it must only be a member of either set A or of set B. The definition of the disjunction “or” also requires that only one statement p or q be true. In set theory there is also a similar term to that of negation in logic. www.cis.yale.edu /ynhti/curriculum/units/1980/7/80.07.04.x.html   (3651 words)

 1.1. Notation and Set Theory   (Site not responding. Last check: 2007-11-06) Sets are the most basic building blocks in mathematics, and it is in fact not easy to give a precise definition of the mathematical object set. B: A union B is the set of all elements that are either in A or in B or in both. B: A intersection B is the set of all elements that are in both sets A and B. web01.shu.edu /projects/reals/logic/notation.html   (1051 words)

 Set Theory Set Theory and Venn Diagrams are very useful for clarifying and understanding classifications and definitions around the form of 'This is an X, it is not a Y' or 'X and Y have some things in common'. In set theory, this means identifying to which sets it belongs. Sets can also be nested, so one set is completely within another set (and is hence a subset). changingminds.org /techniques/argument/set_theory.htm   (490 words)

 03E: Set theory Fuzzy set theory replaces the two-valued set-membership function with a real-valued function, that is, membership is treated as a probability, or as a degree of truthfulness. The theory of finite sets is, arguably, a definition of Combinatorics. Since Axiomatic Set Theory is often used to construct the natural numbers (satisfying the Peano axioms, say) it is possible to translate statements about Number Theory to Set Theory. www.math.niu.edu /~rusin/known-math/index/03EXX.html   (1585 words)

 Set Theory are not since the former set is a set of four objects, while the latter set is a set with only three objects, one of which itself is a set. Note that the empty set is a member of the universal set; it is also a subset of the universal set. This is read as "the set of all pairs {a, b} such that a is an element of the set A and b is an element of the set B". www.rwc.uc.edu /koehler/comath/26.html   (1557 words)

 Set Theory The concepts of sets and set theory is of fundamental importance to the study of mathematics and technology applications, especially in the area of data base structures. Two sets are equal if and only if they contain exactly the same elements, regardless of the order of the elements. The cardinal number of set A, is symbolically represented by n(A), and is read "n of A." Two sets are equivalent if their cardinal numbers are equal. instruction.blackhawk.tec.wi.us /jbellman/sets.htm   (344 words)

 Set Theory The notion `set' is a refinement on the naïve notion `collection': the aim is to put enough constraints on what may be a `set' that certain definitions can make sense and not lead to paradoxes. My main requirements of the notion `set' are that: the collection of natural numbers is a set; the ordinals one obtains from this notion of `set' have certain structural properties that resemble those of the natural numbers; the collection of these ordinals is not a set. Proving that the collection of finite sets obeys its analogue of this looks like it's going to be an interesting exercise, probably depending on the relationship between the finite collections and the natural numbers. www.chaos.org.uk /~eddy/math/set.html   (819 words)

 Amazon.com: Books: Axiomatic Set Theory   (Site not responding. Last check: 2007-11-06) Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning. The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals. In particular, he develops the theory of cardinals by means of a temporary axiom to the effect that equipollent sets have identical cardinalities. www.amazon.com /exec/obidos/tg/detail/-/0486616304?v=glance   (2696 words)

 The Math Forum - Math Library - Set Theory   (Site not responding. Last check: 2007-11-06) A tutorial on sets, convering the definition of sets and their elements, union, intersection, subsets, and sets of numbers. Research in the group is concentrated on axiomatic set theory, in particular: Inner models and large cardinals; Descriptive set theory and Determinacy; Consistency strengths; and Forcing. Set theory was introduced by W. Quine in 1937. forum.swarthmore.edu /library/topics/set_theory   (2290 words)

 Set Theory Each object is known as a member of the set. The universal set is the set of all sets. The set A is a subset of the universal set and so it is within the rectangle. www.mathsrevision.net /alevel/pure/set_theory.php   (286 words)

 Set Theory: Cantor Cantor's last two papers on set theory, Contributions to the foundations of infinite set theory, 1895/1897, give his most polished study of cardinal and ordinal numbers and their arithmetic. By the end of the nineteenth century Cantor was aware of the paradoxes one could encounter in his set theory, e.g., the set of everything thinkable leads to contradictions, as well as the set of all cardinals and the set of all ordinals. The first textbook explicitly devoted to the subject of Cantor's set theory was published in 1906 in England by the Youngs, a famous husband and wife team. www.math.uwaterloo.ca /~snburris/htdocs/scav/cantor/cantor.html   (1069 words)

 Set Theory   (Site not responding. Last check: 2007-11-06) The "cloud-capped V of infinitistic set theory" is the cumulative hierarchy of sets, starting traditionally with the empty set and built up (through systematic application of the axioms of set theory) to higher and higher orders of infinity. First, the axioms for Zermelo-Fraenkel set theory with the axiom of choice and non-set "atoms" or ur-elements are shown in full, since the version accommodating ur-elements is not found in most set theory texts. The class V of all the sets of ZFC/AFA is thus expanded to included non-well-founded sets, i.e., sets which have themselves as members or have members which are related to them by a circular chain of in's (see below). www.greenshade.com /sets.html   (849 words)

 Infinite Ink: The Continuum Hypothesis by Nancy McGough Like the Axiom of Choice (AC), Gödel showed that CH is consistent with standard set theory and Cohen showed that ~CH is consistent with standard set theory (and thus CH is independent of standard set theory). The set of reals between 0 and 1 can be represented by the set of all countably infinite sequences of 0's and 1's. Since CH is not a standard assumption in mathematics and, in fact, most set theorists think it is false, it is important for a writer to state her assumptions about CH. www.ii.com /math/ch   (4563 words)

 Union   (Site not responding. Last check: 2007-11-06) A Union is a single entity which is a collection of two or more entityentities/. MONTREAL (CP) - Just as negotiations between a major teachers' union and the Quebec government appeared to be making progress, talks suddenly broke off Thursday with both sides accusing the other of bad faith. TORONTO (AP) -- A union representing 1,300 minor league hockey players threatened to file an antitrust lawsuit against the NHL over a provision in the league's new collective bargaining agreement, the Toronto Sun reported Thursday. www.infothis.com /find/Union   (396 words)

 Union   (Site not responding. Last check: 2007-11-06) A trade union (also known as a labor union or labour union) is a workers' organization designed to assist in employment negotiation. Union of India - was the official name of India before it became a republic. Union is a 1991 album by the group Yes www.worldhistory.com /wiki/U/Union.htm   (238 words)

 [No title]   (Site not responding. Last check: 2007-11-06) Also, set constraints on their own are extensively studied as a natural formalism for many problems that arise in program analysis (e.g., type-checking or optimization). Non-determinism in set unification, set constraints, intensional set formers, are all features that potentially allow one to write programs in a more declarative fashion, and definitively to obtain simpler and more readable programs. Aim of these pages is to provide a reference point to researchers and practitioners interested in computational uses of set theory. www.cs.nmsu.edu /~complog/sets   (405 words)

 Axiom of Choice   (Site not responding. Last check: 2007-11-06) Given any two sets, one set has cardinality less than or equal to that of the other set -- i.e., one set is in one-to-one correspondence with some subset of the other. In the setting of ordinary set theory, all three of those principles are mathematically equivalent -- i.e., if we assume any one of those principles, we can use it to prove the other two. Thus, it is an alternate form of set theory or higher-order logic, a little different from conventional set theory, but still capable of doing approximately the same things. math.vanderbilt.edu /~schectex/ccc/choice.html   (3751 words)

 Set Theory:Set Operations - Wikibooks We can define the union of two sets A and B using the axioms of union and pair. We define the intersection of two sets using the axiom of separation. We define the difference of two sets using the axiom of separation. en.wikibooks.org /wiki/Set_Theory:Set_Operations   (108 words)

 Union (set theory) : Set theoretic union   (Site not responding. Last check: 2007-11-06) The close, constant, uninterrupted companionship of the married pair have remained forever concealed in city life. Here each was everything blissful consciousness of inseparable union which usually appears only others who were dear to her, but the letters which arrived from time to whenever Pyrrhus went to market, letters reached the island delivered at old freedman, who had become her close friend. So the time came when Dion could say without self-deception that Barine supplied the place of the exciting, changeful life of the capital. www.termsdefined.net /se/set-theoretic-union.html   (929 words)

 Set Theory Handout for Lecture 1: A Short History of Set Theory. This course is an introduction to the fundamentals of set theory. Evaluation will be based upon the mid-term (20%), the final (30%) and the assignments to be set throughout the term (50%). www.nyu.edu /gsas/dept/philo/courses/settheory   (419 words)

 Synopses of Topics - Set Theory The union of two sets is the set of elements that are in at least one of the two sets. A set is drawn as a geometric area (e.g. circle, rectangle) and shading is used to indicate a specific portion of the set or sets. math.usask.ca /emr/sett.html   (502 words)

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