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# Topic: Intersection set theory

 PlanetMath: set theory Set theory is special among mathematical theories, in two ways: It plays a central role in putting mathematics on a reliable axiomatic foundation, and it provides the basic language and apparatus in which most of mathematics is expressed. A category is not a set, and a functor is not a mapping, despite similarities in both cases. This is version 8 of set theory, born on 2003-01-01, modified 2003-02-07. planetmath.org /encyclopedia/SetTheory.html   (940 words)

 Intersection (set theory) - Wikipedia, the free encyclopedia In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. For example, the intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. en.wikipedia.org /wiki/Intersection_(set_theory)   (662 words)

 Set article - Set mathematics mathematics theory 19th century Naive theory Axiomatic - What-Means.com   (Site not responding. Last check: 2007-09-17) 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. The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. www.what-means.com /encyclopedia/Set   (867 words)

 Set - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-09-17) Sets are one of the most important and fundamental concepts in modern mathematics. A set can also have an infinite number of members; for example, the set of natural numbers is infinite. The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = {{unicode∅}}, then A and B are said to be disjoint. encyclopedia.worldsearch.com /set.htm   (1519 words)

 Complement (set theory) - Wikipedia, the free encyclopedia In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A. For example, if the universal set is the set of natural numbers, then the complement of the set of odd numbers is the set of even numbers. en.wikipedia.org /wiki/Set_theoretic_complement   (251 words)

 Union (set theory) 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.brainyencyclopedia.com /encyclopedia/u/un/union__set_theory_.html   (683 words)

 Set theoretic intersection : Intersection (set theory)   (Site not responding. Last check: 2007-09-17) For example, the intersection of the sets {1,2,3} and {2,3,4} is {2,3}. The number 9 is not contained in the intersection of the set of prime numbers {2,3,5,7,11,...} and the set of odd numbers {1,3,5,7,9,11,...}. Intersection is an associative operation; thus, A ∩ (B ∩ C) = (A ∩ B) ∩ C. www.city-search.org /in/intersection-(set-theory).html   (464 words)

 Science Fair Projects - Set The informality of this 'definition' of a set leaves clear that different sets are different; so the definition of a set goes hand in hand with a classification of its objects. So denotes the set of all rational numbers (that is, the set of all proper and improper fractions). This set includes all rational numbers, together with all irrational numbers (that is, numbers which can't be rewritten as fractions, such as π, –π and √2). www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Set   (1547 words)

 set - a Whatis.com definition - see also: set theory 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 Using the SET® Game. The easiest way to think of union is that for any two sets, their union includes all of the elements that are in one or both of the sets. The intersection of two sets is the elements that are in both sets, or the elements the two sets have in common. The complement of a particular set is simply all the elements in the universal set that are not in that set. www.setgame.com /set/set_theory.htm   (836 words)

 PlanetMath: von Neumann-Bernays-Gödel set theory This theory is essentially stronger than ZFC or NBG, as it can prove their consistency (in addition to everything they already prove). by means of the Burali-Forti paradox, that the class of all ordinals is not a set, and hence there is a bijection between the class of ordinals and the class of all sets. This is version 12 of von Neumann-Bernays-Gödel set theory, born on 2003-06-25, modified 2004-09-22. planetmath.org /encyclopedia/VonNeumannBernausGodelSetTheory.html   (796 words)

 Articles - Naive set theory   (Site not responding. Last check: 2007-09-17) Naive set theory was created at the end of the 19th century by Georg Cantor in order to allow mathematicians to work with infinite sets consistently. Two sets A and B are defined to be equal when they have precisely the same elements, that is, if every element of A is an element of B and every element of B is an element of A. The intersection of A and B is the set of all objects which are both in A and in B. www.gaple.com /articles/Naive_set_theory   (2455 words)

 Discrete mathematics:Naive set theory - Wikibooks Sets can be sets of sets as well (bags with bags in them). The intersection of two sets A and B are the elements common to both sets. Given a set A in a larger universal set E, we define the complement of A to be all elements in E that are not in A, that is the complement of A is: en.wikibooks.org /wiki/Discrete_mathematics:Naive_set_theory   (980 words)

 1.1. Notation and Set Theory   (Site not responding. Last check: 2007-09-17) 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   (Site not responding. Last check: 2007-09-17) 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 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. When both sets are seen as a single set, this is called the intersection. changingminds.org /techniques/argument/set_theory.htm   (490 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 intersection says that in order for an element to be part of the solution set for A ½ B it must be both an element of A and an element B. The definition of the conjunction “and” also requires that both statements, p and q be true. 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. www.cis.yale.edu /ynhti/curriculum/units/1980/7/80.07.04.x.html   (3651 words)

 Set Theory   (Site not responding. Last check: 2007-09-17) Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. www.iit.edu /~voraati/cs561   (69 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)

 NTU Info Centre: Intersection (set theory)   (Site not responding. Last check: 2007-09-17) The intersection of A and B is written "A ∩B". For example, the intersection of the sets {1, 2, 3and {2, 3, 4is {2, 3 The number 9 is not contained in the intersection of the set of prime numbers If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩B = Ø. www.nowtryus.com /article:Intersection_(set_theory)   (436 words)

 Encyclopedia article on Set [EncycloZine]   (Site not responding. Last check: 2007-09-17) A set may be defined by specifying in words the property which characterizes it, and enclosing this description in curly braces. F is the set of numbers of the form $n^2– 4, such that n is a whole number between 0 and 19.$ Such a set is called the empty set (or the null set) and is denoted by the symbol Template:Unicode. encyclozine.com /Set   (1243 words)

 Set   (Site not responding. Last check: 2007-09-17) The notion of a set is one of the most important and fundamental concepts in modern mathematics. denotes the set of all rational numbers (that is, the set of all proper and improper fractions). If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E is O, or equivalently, E ′ = O. www.worldhistory.com /wiki/S/Set.htm   (1272 words)

 Set Theory Papers of Andreas R. Blass This survey of the theory of cardinal characteristics of the continuum is to appear as a chapter in the "Handbook of Set Theory." As the title indicates, I concentrate on the combinatorial characteristics; Tomek Bartoszynski has written a chapter on the category and measure characteristics. We apply the Baire category theorem and other classical results of descriptive set theory to the study of the structure of the group Z^{aleph_0} of infinite sequences of integers and some of its subgroups. In the downward-closed context, the ideal of meager sets is prime and b-complete (where b is the bounding number), while the complementary filter is g-complete (where g is the groupwise density cardinal). www.math.lsa.umich.edu /~ablass/set.html   (2059 words)

 The Math Forum - Math Library - Set Theory   (Site not responding. Last check: 2007-09-17) 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. mathforum.org /library/topics/set_theory   (2290 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)

 About "Set Theory" Naive set theory considers elementary properties of the union and intersection operators - Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Axiomatic Set Theory studies the axioms used to describe sets. mathforum.org /library/view/7582.html   (228 words)

 [No title] In the practice of computer science, on the other hand, the use of sets as a data structure is not so common as it might be. Certainly, this is one of the reasons sets are relatively uncommon as data objects. 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). www.cs.nmsu.edu /~complog/sets   (405 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)

 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)

 Zermelo-Frankel Set Theory Thus the set that the Axiom of Infinity declares to exist is the set: The idea is that if P(x,y) corresponds to a function from members of a given set to objects, then we can form a new set by replacing every member of the given set with the object the function maps it to. A somewhat unsettling set.) But it violates the Axiom of Regularity, since its intersection with its only member is not empty. www.trinity.edu /cbrown/topics_in_logic/sets/node4.html   (514 words)

 Integrated generic resource: Clasification and set theory: Scope   (Site not responding. Last check: 2007-09-17) This part of ISO 10303 specifies the existance of a class or set, a classification relationship between a class or set and a member, and set theory relationships between classes or sets. the distinction between a class or set that is defined by abstract criteria, and a class or set that is defined by enumerating its members; relationships that are not classification or set theory relationships. www.cedarlon.demon.co.uk /5_/61/scope.html   (121 words)

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