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International Journal of Research (IJR) Vol-1, Issue-10 November 2014 ISSN 2348-6848 Study on Sets Sujeet Kumar & Ashish Kumar Gupta Department of Information and technology Dronacharya College of Engineering,Gurgaon-122001, India Email:[email protected], Email:[email protected] Set difference of U and A, denoted U \ A, Abstract- is the set of all members of Uthat are not members of A. The set difference {1,2,3} \ Set theory is the branch of mathematical logic that studies sets, which {2,3,4} is {1} , while, conversely, the set are collections of objects. Although any type of object can be collected difference {2,3,4} \ {1,2,3} is {4} . into a set, set theory is applied most often to objects that are relevant to mathematics.In this research paper we studied about Basic concepts and When A is a subset of U, the set notation, some ontology and applications. We have also study about difference U \ A is also called combinational set theory, forcing, cardinal invariants, fuzzy set theory. the complement of A inU. In this case, if We have described all the basic concepts of Set Theory. the choice of U is clear from the context, the notation Acis sometimes used instead Keywords- of U \ A, particularly if U is a universal set as in the study of Venn diagrams. Combinational ;fuzzy ; forcing; cardinals; ontology 1. INTRODUCTION Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all Set theory is the branch of mathematical logic that studies objects that are a member of exactly one sets, which are collections of objects. Although any type of of A and B (elements which are in one of object can be collected into a set, set theory is applied most the sets, but not in both). For instance, for often to objects that are relevant to mathematics. The the sets{1,2,3} and {2,3,4} , the language of set theory can be used in the definitions of symmetric difference set is {1,4} . It is the nearly all mathematical objects. set difference of the union and the intersection, (A ∪ B) \ (A ∩ B) or(A \ B) ∪ 1.1 HISTORY (B \ A). Since the 5th century BC, beginning with Greek Cartesianproduct of A and B, mathematician Zeno of Elea in the West and early Indian denoted A × B, is the set whose members mathematicians in the East, mathematicians had struggled are all possible ordered with the concept of infinity. Especially notable is the work pairs (a,b) where a is a member of Bernard Bolzano in the first half of the 19th century.[3] of A and b is a member of B. The cartesian Modern understanding of infinity began in 1867–71, with product of{1, 2} and {red, white} is {(1, Cantor's work on number theory. An 1872 meeting between red), (1, white), (2, red), (2, white)}. Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Power set of a set A is the set whose members are all possible subsets ofA. For 1.2 BASIC CONCEPTS example, the power set of {1, 2} is { {}, Union of the sets A and B, denoted A ∪ B, {1}, {2}, {1,2} } . is the set of all objects that are a member 1.3 SOME ONTOLOGY of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} . A set is pure if all of its members are sets, all members of its members are sets, and so on. Intersection of the sets A and B, For example, the set {{}} containing only the denoted A ∩ B, is the set of all objects that empty set is a nonempty pure set. In modern are members of both A and B. The set theory, it is common to restrict attention to intersection of {1, 2, 3} and{2, 3, 4} is the the von Neumann universe of pure sets, and set {2, 3} . Study on Sets Sujeet Kumar & Ashish Kumar Gupta P a g e | 1792 International Journal of Research (IJR) Vol-1, Issue-10 November 2014 ISSN 2348-6848 many systems of axiomatic set theory are Kripke–Platek set theory, which designed to axiomatize the pure sets only. omits the axioms of There are many technical advantages to this infinity, powerset, and choice, restriction, and little generality is lost, because and weakens the axiom schemata essentially all mathematical concepts can be of separation and replacement. modeled by pure sets. Sets in the von Neumann universe are organized into a Sets and proper classes. These cumulative hierarchy, based on how deeply include Von Neumann–Bernays–Gödel their members, members of members, etc. are set theory, which has the same strength nested. Each set in this hierarchy is assigned as ZFC for theorems about sets alone, (by transfinite recursion) an ordinal number α, and Morse-Kelley set theory and Tarski– known as its rank. The rank of a pure set X is Grothendieck set theory, both of which defined to be the least upper bound of all are stronger than ZFC. successors of ranks of members of X. For 1.5 APPLICATIONS example, the empty set is assigned rank 0, while the set {{}} containing only the empty Many mathematical concepts can be defined set is assigned rank 1. For each ordinal α, the precisely using only set theoretic concepts. For set Vα is defined to consist of all pure sets example, mathematical structures as diverse with rank less than α. The entire von Neumann as graphs, manifolds, rings, and vector universe is denoted V. spaces can all be defined as sets satisfying various (axiomatic) 1.4 AXIOMATRIC SET THEORY properties. Equivalence and order relations are Elementary set theory can be studied ubiquitous in mathematics, and the theory of informally and intuitively, and so can be mathematical relations can be described in set taught in primary schools using Venn theory. diagrams. The intuitive approach tacitly Set theory is also a promising foundational assumes that a set may be formed from the system for much of mathematics. Since the class of all objects satisfying any particular publication of the first volume of Principia defining condition. This assumption gives rise Mathematica, it has been claimed that most or to paradoxes, the simplest and best known of even all mathematical theorems can be derived which are Russell's paradox and the Burali- using an aptly designed set of axioms for set Forti paradox. Axiomatic set theory was theory, augmented with many definitions, originally devised to rid set theory of such using first or second order logic. For example, paradoxes. properties of the natural and real numbers can The most widely studied systems of axiomatic be derived within set theory, as each number set theory imply that all sets form a cumulative system can be identified with a set hierarchy. Such systems come in two flavors, of equivalence classes under a those whose ontologyconsists of: suitable equivalence relation whose field is some infinite set. Sets alone. This includes the most common axiomatic set theory, Zermelo– Set theory as a foundation for mathematical Fraenkel set theory (ZFC), which analysis, topology, abstract algebra, includes the axiom of choice. Fragments and discrete mathematics is likewise of ZFC include: uncontroversial; mathematicians accept that (in principle) theorems in these areas can be Zermelo set theory, which derived from the relevant definitions and the replaces the axiom schema of axioms of set theory. Few full derivations of replacement with that complex mathematical theorems from set of separation; theory have been formally verified, however, because such formal derivations are often General set theory, a small much longer than the natural language proofs fragment of Zermelo set mathematicians commonly present. One theory sufficient for the Peano verification project, Metamath, includes axioms and finite sets; Study on Sets Sujeet Kumar & Ashish Kumar Gupta P a g e | 1793 International Journal of Research (IJR) Vol-1, Issue-10 November 2014 ISSN 2348-6848 derivations of more than 10,000 theorems In fuzzy set theory this condition was relaxed starting from the ZFC axioms and using first by Lotfi A. Zadeh so an object has a degree of order logic. membership in a set, a number between 0 and 1. For example, the degree of membership of a 1.6 AREAS OF STUDY person in the set of "tall people" is more Combinatorial set theory flexible than a simple yes or no answer and can be a real number such as 0.75. Main article: Infinitarycombinatorics Inner model theory[edit] Combinatorial set theory concerns extensions of finite combinatorics to infinite Main article: Inner model theory sets. This includes the study of cardinal An inner model of Zermelo–Fraenkel set arithmetic and the study of extensions theory (ZF) is a transitive class that includes of Ramsey's theorem such as the Erdős–Rado all the ordinals and satisfies all the axioms of theorem. ZF. The canonical example is the constructible Descriptive set theory universe L developed by Gödel. One reason that the study of inner models is of interest is Main article: Descriptive set theory that it can be used to prove consistency results. For example, it can be shown that regardless Descriptive set theory is the study of subsets of whether a model V of ZF satisfies of the real line and, more generally, subsets the continuum hypothesisor the axiom of of Polish spaces. It begins with the study choice, the inner model L constructed inside of pointclasses in the Borel hierarchy and the original model will satisfy both the extends to the study of more complex generalized continuum hypothesis and the hierarchies such as the projective axiom of choice.
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