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# Topic: Basic set theory

###### In the News (Tue 23 Apr 19)

 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)

 Set theory - Wikipedia, the free encyclopedia In axiomatic set theory, the concepts of sets and set membership are defined indirectly by first postulating certain axioms which specify their properties. Naive set theory is the original set theory developed by mathematicians at the end of the 19th century. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory. en.wikipedia.org /wiki/Set_theory   (375 words)

 Class (set theory) - Wikipedia, the free encyclopedia In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets). A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. en.wikipedia.org /wiki/Class_(set_theory)   (345 words)

 Naïve set theory   (Site not responding. Last check: 2007-10-19) Naïve set theory is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies certain axioms. Sets are of great importance in mathematics; in fact, in the modern formal treatment, the whole machinery of pure mathematics (numbers, relationss, functionss, etc.) is defined in terms of sets. Naïve set theory was developed at the end of the 19th century (principally by Georg Cantor and Gottlob Frege) in order to allow mathematicians to work with infinite sets consistently. www.sciencedaily.com /encyclopedia/naive_set_theory   (1614 words)

 PlanetMath: set theory   (Site not responding. Last check: 2007-10-19) 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)

 Set article - Set mathematics mathematics theory 19th century Naive theory Axiomatic - What-Means.com   (Site not responding. Last check: 2007-10-19) 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.what-means.com /encyclopedia/Set   (867 words)

 Set - Wikipedia Sets are one of the base concepts of mathematics. If A and B are two sets and every x in A is also contained in B, then A is said to be a subset of B. Statistical Theory is built on the base of Set Theory and Probability Theory. nostalgia.wikipedia.org /wiki/Set   (503 words)

 Set Theory (Stanford Encyclopedia of Philosophy) 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. There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe. One of the basic principles of set theory is the existence of an infinite set. plato.stanford.edu /entries/set-theory   (3292 words)

 BASIC SET THEORY   (Site not responding. Last check: 2007-10-19) Naive set theory is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies the axioms. Sets are of great importance in mathematics; in fact, in the modern formal treatment, the whole machinery of pure mathematics (numbers, relations, functions, etc.) is defined in terms of sets. Naive set theory was developed at the end of the 19th century (principally by Georg Cantor and Frege) in order to allow mathematicians to work with infinite sets consistently. www.websters-online-dictionary.org /definition/BASIC+SET+THEORY   (1521 words)

 Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy) An infinite subset of a countable set is countable. The set of all finite subsets of a countable set is countable. The set of all integers Z and the set of all rational numbers Q are countable. plato.stanford.edu /entries/set-theory/primer.html   (2793 words)

 Talk:Naive set theory : Talk:Basic Set Theory   (Site not responding. Last check: 2007-10-19) I think what was meant was the set theory article, but nobody seems to be brave enough to refactor it. First, ZFC isn't all that new compared to set theory in general, as a mathematical concept; that is, we've had ZFC for most of the time that we've had sets, by now. Basic set theory is about what mathematicians call "naive set theory", while Set theory is about what we call "axiomatic set theory", and I argue that the articles should have those names. www.termsdefined.net /ta/talk:basic-set-theory.html   (916 words)

 Basic Set Theory/Talk - Wikipedia Both this and the set article (whose title is very ambiguous) are needed for different levels of study. Thats what I meant, and I also did not mean to disparage this one, rather that the two should be merged, with the strengths of this one added to the old, rather than presented as an alternative. Basic Set Theory should remain as a separate article. nostalgia.wikipedia.org /wiki/Basic_Set_Theory/Talk   (181 words)

 Set Theory Basic set theory used to be the place where all mathematics courses started from Algebra 1 in high school to graduate level courses. The set A is called the domain of the function and the set B is called the codomain of the function. The codomain of a function f is the set of all elements that could be hit by f; the range is the set of all elements which are hit. www.iwu.edu /~lstout/theoremlist/node5.html   (486 words)

 set - a Whatis.com definition - see also: set theory   (Site not responding. Last check: 2007-10-19) 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)

 :: Quantnotes.com :: Fundamentals :: This article is intended as an introduction to basic set theory for those aiming to apply it in the field of probability theory and (eventually) stochastic calculus. In basic probability the probability of the union of 2 events A and B is often described by the probability of events A or B being observed. The set {1,3, 5, 7} is a subset of the much larger set of all odd integers but it is not a subset of the set {1,3,5}, since 7 is not a member of the smaller set. www.quantnotes.com /fundamentals/backgroundmaths/settheory.htm   (1434 words)

 1.1. Notation and Set Theory 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)

 Read This: Three Books on Set Theory The three books under review are all concerned with sets in some manner or other, and purport to be introductions to set theory, at least after a fashion. I belong to the former set in that I do adore category theory and believe it to be, among many other good things, a wonderful means for becoming able to see deep structural similarities within mathematics, ignoring even natural borders in sometimes dramatic ways. Basic Set Theory, by A. Shen and N. Verschagin. www.maa.org /reviews/setbooks3.html   (1340 words)

 Set theory, analysis and their neighbours, Second Meeting Finite model theory and set theory are a priori distinct, the former dealing with finite and the latter with infinite structures. However beyond this one is immediately struck by the basic role of abstract (as opposed to applied) model theory and by the importance of extensions of first order logic in each. Set theory, analysis and their neighbours (the first meeting in the series), inlcuding slides from the talks and related preprints. www.ucl.ac.uk /~ucahcjm/stn/stn2.html   (594 words)

 Basic Set Theory Set theory lies at the foundation of all modern mathematics. It was realized at the turn of this century that not every collection of elements can be allowed to constitute a set without leading to paradoxes. These are formulated in various axiom systems for the theory of sets. www.mathresource.iitb.ac.in /project/index.html   (115 words)

 sciforums.com - Set Theory Set theory appears in quite a different number of places in advanced mathematics (like measure theory), but indeed you need to get familiar with the basics before you can reason in more advanced and abstract settings. Set theory is the very foundation of permutations and combinations, without those rules that bound set theory together the above problem would be very difficult indeed!! Set theory gives you a nice framework that cuts down the sample space making probability questions a breeze. www.sciforums.com /showthread.php?t=25297   (976 words)

 Additional Reading (from set theory) --  Encyclopædia Britannica The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. In those societies chiefly identified with the practice, a person belonged, either from birth or from a determined age, to a named age set that passed through a series of stages, each of which had a distinctive status or social and political role. in mathematics and mechanics, theory that studies systems behaving unpredictably and randomly despite their seeming simplicity and fact that forces involved are supposedly governed by well-understood physical laws; applications of theory are diverse, including study of turbulent flow of fluids, irregularities in heartbeat, traffic jams, population dynamics, chemical... www.britannica.com /eb/article-24046?tocId=24046   (1235 words)

 Oz's crib sheet: basic set theory   (Site not responding. Last check: 2007-10-19) While an ordinary set just has elements, a collection has elements' elements (the elements of the sets which are its elements). Operations on families whose index set is 2: If you think of 2 as the set {1, 2}, then 2 is just another possible index set for a family; such a family is a finite sequence of length 2, or ordered pair. Exponentiation of sets: If X and Y are sets, consider the constant family of sets whose index set is Y and whose constant value is X. www.math.ucr.edu /~toby/Oz/sets   (3101 words)

 Math: Sets We often deal with groups or collection of objects, such a set of books, a group of students, a list of states in a country, a collection of baseball cards, etc. Sets may be thought of as a mathematical way to represent collections or groups of objects. The concept of sets is an essential foundation for various other topics in mathematics. This series of lessons cover the essential concepts of math set theory - the basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, basic set operations including intersection and union of sets, using Venn diagrams and simple applications of sets. www.onlinemathlearning.com /math-sets.html   (114 words)

 Citations: Bounded set theory and polynomial computability - Yu (ResearchIndex) The class of Delta definable operations of the theory BSTC in the universe of hereditarily finite sets coincides with P = P fl : Delta[BSTC] HF P : Another natural tree encoding of HF is the restriction of graph encoding fl to the subclass T AG of finite trees. Sets which may be axiomatized (of course, non categorically) without any reference to the powerset operation. A Bounded Set Theory with Anti-Foundation Axiom and Inductive.. citeseer.ist.psu.edu /context/208654/0   (1275 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. 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. www.cs.nmsu.edu /~complog/sets   (405 words)

 Set Theory Explores sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories. The first part of this advanced-level text covers pure set theory, and the second deals with applications and advanced topics (point set topology, real spaces, Boolean algebras, infinite combinatorics and large cardinals). The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise store.doverpublications.com /by-subject-science-and-mathematics-mathematics-set-theory.html   (441 words)

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