Chapter 1 : Discrete Mathematics/Set theory - Wikibooks, open books for an open world Set theory is a branch of mathematical logic that studies sets, which informally are collections of theinnatdunvilla.comgh any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Otherwise, player II wins. Further, he showed that if there exists a large cardinal called measurable see Section 10, then even the analytic sets are determined. The axiom of Projective Determinacy PD asserts that every projective set is determined. It turns out that PD implies that all projective sets of reals are regular, and Woodin has shown that, in a certain sense, PD settles essentially all questions about the projective sets. Moreover, PD seems to be necessary for this. Thus, the CH holds for closed sets. More than thirty years later, Pavel Aleksandrov extended the result to all Borel sets, and then Mikhail Suslin to all analytic sets. Thus, all analytic sets satisfy the CH. However, the efforts to prove that co-analytic sets satisfy the CH would not succeed, as this is not provable in ZFC. See the entry on the continuum hypothesis for the current status of the problem, including the latest results by Woodin. It is in fact the smallest inner model of ZFC, as any other inner model contains it. The theory of constructible sets owes much to the work of Ronald Jensen. He also proved the consistency of the negation of the AC, relative to the consistency of ZF. To achieve this, Cohen devised a new and extremely powerful technique, called forcing, for expanding transitive models of ZFC. Since all hereditarily-finite sets are constructible, we aim to add an infinite set of natural numbers. Suslin conjectured that this is still true if one relaxes the requirement of containing a countable dense subset to being ccc, i. About the same time, Robert Solovay and Stanley Tennenbaum developed and used for the first time the iterated forcing technique to produce a model where the SH holds, thus showing its independence from ZFC. This is why a forcing iteration is needed. The search for new axioms As a result of 50 years of development of the forcing technique, and its applications to many open problems in mathematics, there are now literally thousands of questions, in practically all areas of mathematics, that have been shown independent of ZFC. These include almost all questions about the structure of uncountable sets. One might say that the undecidability phenomenon is pervasive, to the point that the investigation of the uncountable has been rendered nearly impossible in ZFC alone see however Shelah for remarkable exceptions. This prompts the question about the truth-value of the statements that are undecided by ZFC. Should one be content with them being undecidable? Does it make sense at all to ask for their truth-value? There are several possible reactions to this. See Hauser for a thorough philosophical discussion of the Program, and also the entry on large cardinals and determinacy for philosophical considerations on the justification of new axioms for set theory. A central theme of set theory is thus the search and classification of new axioms. These fall currently into two main types: Thus, the existence of a regular limit cardinal must be postulated as a new axiom. Such a cardinal is called weakly inaccessible. If the GCH holds, then every weakly inaccessible cardinal is strongly inaccessible. Large cardinals are uncountable cardinals satisfying some properties that make them very large, and whose existence cannot be proved in ZFC. The first weakly inaccessible cardinal is just the smallest of all large cardinals. Beyond inaccessible cardinals there is a rich and complex variety of large cardinals, which form a linear hierarchy in terms of consistency strength, and in many cases also in terms of outright implication. See the entry on independence and large cardinals for more details. Much stronger large cardinal notions arise from considering strong reflection properties. A strengthening of this principle to second-order sentences yields some large cardinals. By allowing reflection for more complex second-order, or even higher-order, sentences one obtains large cardinal notions stronger than weak compactness. All known proofs of this result use the Axiom of Choice, and it is an outstanding important question if the axiom is necessary. Another important, and much stronger large cardinal notion is supercompactness. Woodin cardinals fall between strong and supercompact. Beyond supercompact cardinals we find the extendible cardinals, the huge, the super huge, etc. The strongest large cardinal notions not known to be inconsistent, modulo ZFC, are the following: Large cardinals form a linear hierarchy of increasing consistency strength. In fact they are the stepping stones of the interpretability hierarchy of mathematical theories. As we already pointed out, one Page 1
cannot prove in ZFC that large cardinals exist. But everything indicates that their existence not only cannot be disproved, but in fact the assumption of their existence is a very reasonable axiom of set theory. For one thing, there is a lot of evidence for their consistency, especially for those large cardinals for which it is possible to construct an inner model. For instance, it has a projective well ordering of the reals, and it satisfies the GCH. Further, under a weaker large-cardinal hypothesis, namely the existence of infinitely many Woodin cardinals, Martin and Steel proved that every projective set of real numbers is determined, i. He also showed that Woodin cardinals provide the optimal large cardinal assumptions by proving that the following two statements: There are infinitely many Woodin cardinals. See the entry on large cardinals and determinacy for more details and related results. Another area in which large cardinals play an important role is the exponentiation of singular cardinals. The so-called Singular Cardinal Hypothesis SCH completely determines the behavior of the exponentiation for singular cardinals, modulo the exponentiation for regular cardinals. The SCH holds above the first supercompact cardinal Solovay. Large cardinals stronger than measurable are actually needed for this. Moreover, if the SCH holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals Silver. MA asserts the following: At first sight, MA may not look like an axiom, namely an obvious, or at least reasonable, assertion about sets, but rather like a technical statement about ccc partial orderings. It does look more natural, however, when expressed in topological terms, for it is simply a generalization of the well-known Baire Category Theorem, which asserts that in every compact Hausdorff topological space the intersection of countably-many dense open sets is non-empty. Indeed, MA is equivalent to: MA has many different equivalent formulations and has been used very successfully to settle a large number of open problems in other areas of mathematics. See Fremlin for many more consequences of MA and other equivalent formulations. In spite of this, the status of MA as an axiom of set theory is still unclear. Perhaps the most natural formulation of MA, from a foundational point of view, is in terms of reflection. Writing HC for the set of hereditarily-countable sets i. Much stronger forcing axioms than MA were introduced in the s, such as J. The PFA asserts the same as MA, but for partial orderings that have a property weaker than the ccc, called properness, introduced by Shelah. Strong forcing axioms, such as the PFA and MM imply that all projective sets of reals are determined PD, and have many other strong consequences in infinite combinatorics. Springer Monographs in Mathematics, New York: Reprinted in Zermelo English translation also in van Heijenoort Page 2
Chapter 2 : GMAT Set theory Practice Questions Sets Sample Questions Wizako Online GMAT classes In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x √ A, while x ∠A indicates that x is not a member of A. Sets and proper classes. The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members. NF and NFU include a "set of everything, " relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject. Applications[ edit ] Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various axiomatic properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory. Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set. Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that in principle theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. Areas of study[ edit ] Set theory is a major area of research in mathematics, with many interrelated subfields. Combinatorial set theory[ edit ] Main article: Infinitary combinatorics Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. Descriptive set theory[ edit ] Main article: Descriptive set theory Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals. The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending "relativizing" it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics. Fuzzy set theory[ edit ] Main article: Fuzzy set theory In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0. Inner model theory[ edit ] Main article: Inner model theory An inner model of Zermeloâ Fraenkel set theory ZF is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum Page 3
hypothesis and the axiom of choice. Thus the assumption that ZF is consistent has at least one model implies that ZF together with these two principles is consistent. The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy and thus not satisfying the axiom of choice. Large cardinal property A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory. Determinacy Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy AD is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved in particular, measurable and with the perfect set property. AD can be used to prove that the Wadge degrees have an elegant structure. Forcing mathematics Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined i. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models. Cardinal invariant A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality 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 Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo-Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]". He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results. Page 4
Chapter 3 : GMAT Quantitative: Formulas for Set Theory - Kaplan Test Prep Set symbols of set theory and probability with name and definition: set, subset, union, intersection, element, cardinality, empty set, natural/real/complex number set. A Venn diagram can be a useful way of illustrating relationships between sets. In a Venn diagram: The universal set is represented by a rectangle. Points inside the rectangle represent elements that are in the universal set; points outside represent things not in the universal set. Other sets are represented by loops, usually oval or circular in shape, drawn inside the rectangle. Again, points inside a given loop represent elements in the set it represents; points outside represent things not in the set. On the right A is a subset of B, because the loop representing set A is entirely enclosed by loop B. Worked Examples[ edit ] Venn diagrams: Note that the rectangle representing the universal set is divided into four regions, labelled i, ii, iii and iv. What can be said about the sets A and B if it turns out that: So A is a subset of B, and the diagram should be re-drawn like Fig 2 above. The diagram should then be re-drawn like Fig 1 above. Example 2 a Draw a Venn diagram to represent three sets A, B and C, in the general case where nothing is known about possible relationships between the sets. In each case, the Venn diagram can be re-drawn so that empty regions are no longer included. If region ii is empty, the loop representing A should be made smaller, and moved inside B and C to eliminate region ii. If regions ii, iii and iv are empty, make A and B smaller, and move them so that they are both inside C thus eliminating all three of these regions, but do so in such a way that they still overlap each other thus retaining region vi. Drawing Venn diagrams for each of the above examples is left as an exercise for the reader. Example 3 The following sets are defined: The technique is as follows: Go through the elements of the universal set one at a time, once only, entering each one into the appropriate region of the diagram. Re-draw the diagram, if necessary, moving loops inside one another or apart to eliminate any empty regions. Solution After drawing the three empty loops in a diagram looking like Fig. No Are you in B? No Are you in C? So write it in the appropriate region region number i in Fig. Yes Are you in B? Yes Are you in C? No Yes, yes, no: Goes in region iii then. The resulting diagram looks like Fig. So we need to: The finished result is shown in Fig. This gives us a very simple pattern, as follows: With one set loop, there will be just two regions: Each new loop we add to the diagram divides each existing region into two, thus doubling the number of regions altogether. Chapter 4 : Set Theory: theinnatdunvilla.com Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the theinnatdunvilla.com set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. Chapter 5 : Zermeloâ Fraenkel set theory - Wikipedia Set Theory. Showing top 8 worksheets in the category - Set Theory. Some of the worksheets displayed are 07 ws1 sets practice work set theory class vii, Basic set theory, An introduction to set theory, Basic concepts of set theory functions and relations, Set operations, Cxc csec mathematics lesson unit three set theory, Mathematics work sets, Sets of real numbers date period. Chapter 6 : set theory Basics, Examples, & Formulas theinnatdunvilla.com Chapter 2 Basic Set Theory A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical in-. Chapter 7 : Set symbols of set theory (Ã,U,{},∈,) Page 5
A mathematical set is defined as an unordered collection of distinct elements. That is, elements of a set can be listed in any order and elements occurring more than once are equivalent to occurring only once. We say that an element is a member of a set. An element of a set can be anything. It's. Chapter 8 : Set Theory/Sets - Wikibooks, open books for an open world Advanced. Show Ads. Hide Ads About Ads. Chapter 9 : Set Theory Tutorial Problems, Formulas, Examples MBA Crystal Ball Chapter 0 Introduction Set Theory is the true study of inï nity. This alone assures the subject of a place prominent in human culture. But even more, Set Theory is the milieu. Page 6