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]

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} \ is the branch of 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 .In this research paper we studied about Basic concepts and When A is a of U, the set notation, some and applications. We have also study about difference U \ A is also called combinational set theory, , cardinal invariants, theory. the 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 as in the study of Venn diagrams. Combinational ;fuzzy ; forcing; cardinals; ontology

1. INTRODUCTION  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 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 . Especially notable is the work pairs (a,b) where a is a member of 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' work on theory. An 1872 meeting between red), (1, white), (2, red), (2, white)}. Cantor and influenced Cantor's thinking and culminated in Cantor's 1874 paper.  of a set A is the set whose members are all possible 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 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 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 of There are many technical advantages to this infinity, powerset, and choice, restriction, and little generality is lost, because and weakens the schemata essentially all mathematical concepts can be of separation and replacement. modeled by pure sets. Sets in the are organized into a  Sets and proper classes. These , 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 about sets alone, (by transfinite ) an 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 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 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 is some .  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 , includes the . Fragments and discrete mathematics is likewise of ZFC include: uncontroversial; mathematicians accept that (in principle) theorems in these areas can be  , which derived from the relevant definitions and the replaces the 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  , 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 such as 0.75. Main article: Infinitarycombinatorics Inner [edit] Combinatorial set theory concerns extensions of finite combinatorics to infinite Main article: sets. This includes the study of cardinal An inner model of Zermelo–Fraenkel set and the study of extensions theory (ZF) is a transitive class that includes of Ramsey's such as the Erdős–Rado all the ordinals and satisfies all the axioms of theorem. ZF. The canonical example is the constructible 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 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 in the and the original model will satisfy both the extends to the study of more complex generalized and the hierarchies such as the projective axiom of choice. Thus the assumption that ZF hierarchy and the Wadge hierarchy. Many is consistent (has at least one model) implies properties of Borel sets can be established in that ZF together with these two principles is ZFC, but proving these properties hold for consistent. more complicated sets requires additional axioms related to and large The study of inner models is common in the cardinals. study of determinacy and large cardinals, especially when considering axioms such as The field of effective descriptive set theory is the that contradict the between set theory and recursion theory. It axiom of choice. Even if a fixed model of set includes the study of lightface pointclasses, theory satisfies the axiom of choice, it is and is closely related to hyperarithmetical possible for an inner model to fail to satisfy theory. In many cases, results of classical the axiom of choice. For example, the descriptive set theory have effective versions; existence of sufficiently large cardinals in some cases, new results are obtained by implies that there is an inner model satisfying proving the effective version first and then the axiom of determinacy (and thus not extending ("relativizing") it to make it more [5] satisfying the axiom of choice). broadly applicable. Large cardinals A recent area of research concerns Borel equivalence relations and more complicated A is a with an definable equivalence relations. This has extra property. Many such properties are important applications to the study studied, including inaccessible of invariants in many fields of mathematics. cardinals, measurable cardinals, and many more. These properties typically imply the Fuzzy set theory[edit] cardinal number must be very large, with the Main article: Fuzzy set theory existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set In set theory as Cantor defined theory. and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. Determinacy

Study on Sets Sujeet Kumar & Ashish Kumar Gupta P a g e | 1794

International Journal of Research (IJR) Vol-1, Issue-10 November 2014 ISSN 2348-6848

Determinacy refers to the fact that, under Set-theoretic topology studies questions appropriate assumptions, certain two-player of general topology that are set-theoretic in games of perfect information are determined nature or that require advanced methods of set from the start in the sense that one player must theory for their solution. Many of these have a winning strategy. The existence of theorems are independent of ZFC, requiring these strategies has important consequences in stronger axioms for their proof. A famous descriptive set theory, as the assumption that a problem is the normal Moore space question, a broader class of games is determined often question in general topology that was the implies that a broader class of sets will have a subject of intense research. The answer to the topological property. The axiom of normal Moore space question was eventually determinacy (AD) is an important object of proved to be independent of ZFC. study; although incompatible with the axiom of choice, AD implies that all subsets of the 1.7 REFRENCES real line are well behaved (in particular,  Devlin, Keith, 1993. The Joy of Sets (2nd measurable and with the perfect set property). ed.). Springer Verlag, ISBN 0-387-94094- AD can be used to prove that the Wedge 4 degrees have an elegant structure.  Ferreirós, Jose, 2007 (1999). Labyrinth of Forcing[edit] Thought: A history of set theory and its Main article: Forcing (mathematics) role in modern mathematics. Basel, Birkhäuser. ISBN 978-3-7643-8349-7 invented the method of forcing while searching for  Johnson, Philip, 1972. A History of Set a model of ZFC in which the continuum Theory. Prindle, Weber & Schmidt ISBN hypothesis fails, or a model of ZF in which 0-87150-154-6 the axiom of choice fails. Forcing adjoins to  Kunen, Kenneth, 1980. Set Theory: An some given model of set theory additional sets Introduction to Independence Proofs. in order to create a larger model with North-Holland, ISBN 0-444-85401-0. properties determined (i.e. "forced") by the construction and the original model. For  Potter, Michael, 2004. Set Theory and Its example, Cohen's construction adjoins Philosophy: A Critical additional subsets of the natural Introduction. Oxford University Press. numbers without changing any of the cardinal numbers of the original model. Forcing is also  Tiles, Mary, 2004 (1989). The Philosophy one of two methods for proving relative of Set Theory: An Historical Introduction consistency by finitistic methods, the other to Cantor's Paradise. method being Boolean-valued models.

Cardinal invariants[edit]

A cardinal invariant is a property of the real

line measured by a cardinal number. For example, a well-studied invariant is the smallest of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

Set-theoretic topology[edit]

Study on Sets Sujeet Kumar & Ashish Kumar Gupta P a g e | 1795