DOCSLIB.ORG

Set Theory Handout

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Set Theory Handout Salem State College MAT 420 Set Theory L. Pedro Poitevin October 25, 2007 1 Foundations De¯nition (Informal). We de¯ne a set A to be any unordered collection of objects, e.g., f3; 8; 5; 1g is a set. If x is an object, we say that x is an element of A or x 2 A if x lies in the collection; otherwise we say that x is not an element of A and write x2 = A. For instance, 3 2 f1; 2; 3; 4; 5g but 7 2= f1; 2; 3; 4; 5g. Axiom 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is an element of B. De¯nition (Equality of sets). Two sets A and B are equal|in notation A = B|if and only if every element of A is an element of B and vice versa. Axiom 2 (Empty set). There exists a set ;, known as the empty set, which contains no elements, i.e. for every object x we have x2 = ;. Exercise. Prove that there is only one empty set. Lemma 1 (Single choice). Let A be a nonempty set. Then there exists an object x such that x 2 A. Axiom 3 (Singletons). If a is an object, then there exists a set fag whose only element is a. Axiom 4 (Pairwise union). Given any two sets A and B, there exists a set A [ B called the union of A and B, whose elements are all objects which are elements of A or B. In symbols, x 2 A [ B () x 2 A _ x 2 B. Remark. This axiom allows us to de¯ne pair sets, triplet sets, quadruplet sets, and so forth. If a; b are two objects, we de¯ne fa; bg := fag [ fbg; if a; b; c are three objects, then fa; b; cg = fag [ fbg [ fcg, and so on. Lemma 2. The union operation above described is commutative and associative. Remark. Because of the associativity of the union operation, we do not need to use parentheses to denote multiple unions; thus, for instance, we can write A[B [C instead of (A[B)[C or A[(B [C). Similarly for unions of four sets A [ B [ C [ D, etc. De¯nition (Subsets). Let A and B be sets. We say that A is a subset of B and write A ⊆ B if every element of A is an element of B. We say that A is a proper subset of B and write A ½ B if A ⊆ B and A 6= B. Proposition 1 (Sets are partially ordered by set inclusion). Let A; B; C be sets. The following three conditions are satis¯ed: 1. If A ⊆ B and B ⊆ A, then A = B. 2. If A ⊆ B and B ⊆ C, then A ⊆ C. 3. If A ½ B and B ½ C, then A ½ C. Axiom 5 (Speci¯cation). Let A be a set, and for each x 2 A, let P (x) be a property pertaining to x (i.e., P (x) is either a true statement or a false statement). Then there exists a set fx 2 A : P (x) is trueg (or simply fx 2 A : P (x)g for short), whose elements are precisely the elements x in A for which P (x) is true. In symbols, for any object y, y 2 fx 2 A : P (x)g () y 2 A ^ P (y) is true. De¯nition (Pairwise intersection). The intersection S1 \ S2 of two sets S1;S2 is de¯ned to be the set S1 \ S2 := fx 2 S1 : x 2 S2g. In other words, x 2 S1 \ S2 () x 2 S1 ^ x 2 S2. Remark. One should be careful with the English word \and": rather confusingly, it can mean either union or intersection, depending on the context. For instance, if one talks about a set of \boys and girls," one means the union of a set of boys and a set of girls, but if one talks about the set of people who are single and male, then one means the intersection of the set of single people with the set of male people. (Can you work out the rule of grammar that determines when \and" means union and when \and" means intersection?) Another problem is that \and" is used in English to denote addition. So, for instance, one could say that \2 and 3 is 5," while also saying that the elements of f2g and the elements of f3g form the set f2; 3g" and \the elements of f2g and f3g form the set ;." This can certainly get confusing! One reason to use mathematical symbols instead of English words such as \and" is that mathematical symbols almost always have a precise and unambiguous meaning, whereas one must often look very carefully at the context in order to work out what an English word means. De¯nition. Two sets A and B are said to be disjoint if A \ B = ;. Remark. Note that being disjoint is not the same as being distinct. If A and B are disjoint, then they are possibly distinct (¯nd an exception!) but if they are distinct, they may not be disjoint. De¯nition (Set di®erence). Given two sets A and B, we de¯ne the set A n B to be the set fx 2 A : x2 = Bg. Proposition 2 (Sets form a Boolean algebra). Let A; B; C be sets, and let X be a set containing A; B; C as subsets. (a) A [; = A and A \; = ;. (b) A [ X = X and A \ X = A. (c) A \ A = A and A [ A = A. (d) A [ B = B [ A and A \ B = B \ A. (e) (A [ B) [ C = A [ (B [ C) and (A \ B) \ C = A \ (B \ C). (f) A \ (B [ C) = (A \ B) [ (A \ C) and A [ (B \ C) = (A [ B) \ (A [ C) (g) A [ (X n A) = X and A \ (X n A) = ;. (h) X n (A [ B) = (X n A) \ (X n B) and X n (A \ B) = (X n A) [ (X n B). Axiom 6 (Replacement). Let A be a set. For any object x 2 A and any object y, suppose we have a statement P (x; y) pertaining to x and y, such that for each x 2 A there is at most one y for which P (x; y) is true. Then there exists a set fy : P (x; y) is true for some x 2 Ag, such that for any object z, z 2 fy :P (x; y) is true for some x 2 Ag () P (x; z) is true for some x 2 A. Axiom 7 (In¯nity). There exists a set N whose elements are called natural numbers, as well as an object 0 2 N and an object n ++ assigned to every natural number n 2 N, such that the Peano axioms hold. Before the development of modern axiomatic set theory, logicians and mathematicians by and large accepted a single, unifying principle, called universal speci¯cation, which implies most of the axioms above described. This principle is stated as follows: \Suppose for every object x we have a property P (x) pertaining to x. Then there exists a set fx : P (x) is trueg such that for every object y, y 2 fx : P (x) is trueg () P (y) is true." (1) The principle of universal speci¯cation is also known as the axiom of comprehension, and it was at the heart of the foundation of mathematics, as understood by brilliant philosophers and mathematicians at the onset of the XX century, particularly Gottlieb Frege. Unfortunately for Frege, and for many mathematicians of his time, this so-called axiom was erroneous, as the venerable philosopher, mathematician, peace-activist, Nobel-prize winning writer, and fantastic educator Lord Bertrand Russell found out. Bertie Russell, whose exploits make him the grandfather of modern mathematical logic, considered the statement P (x) that says \x is a set, and x2 = x." Such a statement looks innocent enough, and it is easy to verify that most sets satisfy that statement. The exceptions seem problematic, true enough, but he concentrated his attention not on the exceptions but on the innocent looking sets x satisfying P (x). (Example: P (f0; 1g) is true, because f0; 1g is not an element of f0; 1g. On the other hand, P (fx : x is a setg) is presumably not true, because a priori fx : x is a setg is itself a set, and so it is an element of itself.) Bertie used the principle of universal speci¯cation to produce the very large set ­ de¯ned below: ­ := fx : P (x) is trueg = fx : x is a set and x2 = xg, i.e., the set of all sets which do not contain themselves (as elements). Bertie ¯nally asked himself (and all of you) the following mind-boggling question, which e®ectively destroyed Frege's axiomatic edi¯ce for all of mathematics: Question (Russell's paradox). Is ­ 2 ­? The problem with the principle of universal speci¯cation is, in some sense, that it creates sets that are \too large." For example, one can talk about the set of all sets, and so one has some sets that contain themselves as elements, which is not an ideal thing. Now that we have seen just how awful the situation can get pretty easily, we add one axiom below to ensure that we do not get any ugly paradoxes like Russell's paradox above.
Load more

Recommended publications
  • Certain Ideals Related to the Strong Measure Zero Ideal
    数理解析研究所講究録 第 1595 巻 2008 年 37-46 37 Certain ideals related to the strong measure zero ideal 大阪府立大学理学系研究科 大須賀昇 (Noboru Osuga) Graduate school of Science, Osaka Prefecture University 1 Motivation and Basic Definition In 1919, Borel[l] introduced the new class of Lebesgue measure zero sets called strong measure zero sets today. The family of all strong measure zero sets become $\sigma$-ideal and is called the strong measure zero ideal. The four cardinal invariants (the additivity, covering number, uniformity and cofinal- ity) related to the strong measure zero ideal have been studied. In 2002, Yorioka[2] obtained the results about the cofinality of the strong measure zero ideal. In the process, he introduced the ideal $\mathcal{I}_{f}$ for each strictly increas- ing function $f$ on $\omega$ . The ideal $\mathcal{I}_{f}$ relates to the structure of the real line. We are interested in how the cardinal invariants of the ideal $\mathcal{I}_{f}$ behave. $Ma\dot{i}$ly, we te interested in the cardinal invariants of the ideals $\mathcal{I}_{f}$ . In this paper, we deal the consistency problems about the relationship between the cardi- nal invariants of the ideals $\mathcal{I}_{f}$ and the minimam and supremum of cardinal invariants of the ideals $\mathcal{I}_{g}$ for all $g$ . We explain some notation which we use in this paper. Our notation is quite standard. And we refer the reader to [3] and [4] for undefined notation. For sets X and $Y$, we denote by $xY$ the set of all functions $homX$ to Y.
    [Show full text]
  • A Taste of Set Theory for Philosophers
    Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
    [Show full text]
  • COVERING PROPERTIES of IDEALS 1. Introduction We Will Discuss The
    COVERING PROPERTIES OF IDEALS MAREK BALCERZAK, BARNABAS´ FARKAS, AND SZYMON GLA¸B Abstract. M. Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices of the required subsequence should be in a fixed ideal J on !. We introduce the notion of the J-covering property of a pair (A;I) where A is a σ- algebra on a set X and I ⊆ P(X) is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal N and the meager ideal M. We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals J on ! such that M has the J- covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katˇetov-Blass order, in the family of those ideals J on ! such that N or M has the J-covering property. Furthermore, we prove a general result about the cases when the covering property \strongly" fails. 1. Introduction We will discuss the following result due to Elekes [8].
    [Show full text]
  • Product Specifications and Ordering Information CMS 2019 R2 Condition Monitoring Software
    Product specifications and ordering information CMS 2019 R2 Condition Monitoring Software Overview SETPOINT® CMS Condition Monitoring Software provides a powerful, flexible, and comprehensive solution for collection, storage, and visualization of vibration and condition data from VIBROCONTROL 8000 (VC-8000) Machinery Protection System (MPS) racks, allowing trending, diagnostics, and predictive maintenance of monitored machinery. The software can be implemented in three different configurations: 1. CMSPI (PI System) This implementation streams data continuously from all VC-8000 racks into a connected PI Server infrastructure. It consumes on average 12-15 PI tags per connected vibration transducer (refer to 3. manual S1176125). Customers can use their CMSHD/SD (Hard Drive) This implementation is used existing PI ecosystem, or for those without PI, a when a network is not available stand-alone PI server can be deployed as a self- between VC-8000 racks and a contained condition monitoring solution. server, and the embedded flight It provides the most functionality (see Tables 1 recorder in each VC-8000 rack and 2) of the three possible configurations by will be used instead. The flight recorder is a offering full integration with process data, solid-state hard drive local to each VC-8000 rack integrated aero/thermal performance monitoring, that can store anywhere from 1 – 12 months of integrated decision support functionality, nested data*. The data is identical to that which would machine train diagrams and nearly unlimited be streamed to an external server, but remains in visualization capabilities via PI Vision or PI the VC-8000 rack until manually retrieved via ProcessBook, and all of the power of the OSIsoft removable SDHC card media, or by connecting a PI System and its ecosystem of complementary laptop to the rack’s Ethernet CMS port and technologies.
    [Show full text]
  • Contents 3 Homomorphisms, Ideals, and Quotients
    Ring Theory (part 3): Homomorphisms, Ideals, and Quotients (by Evan Dummit, 2018, v. 1.01) Contents 3 Homomorphisms, Ideals, and Quotients 1 3.1 Ring Isomorphisms and Homomorphisms . 1 3.1.1 Ring Isomorphisms . 1 3.1.2 Ring Homomorphisms . 4 3.2 Ideals and Quotient Rings . 7 3.2.1 Ideals . 8 3.2.2 Quotient Rings . 9 3.2.3 Homomorphisms and Quotient Rings . 11 3.3 Properties of Ideals . 13 3.3.1 The Isomorphism Theorems . 13 3.3.2 Generation of Ideals . 14 3.3.3 Maximal and Prime Ideals . 17 3.3.4 The Chinese Remainder Theorem . 20 3.4 Rings of Fractions . 21 3 Homomorphisms, Ideals, and Quotients In this chapter, we will examine some more intricate properties of general rings. We begin with a discussion of isomorphisms, which provide a way of identifying two rings whose structures are identical, and then examine the broader class of ring homomorphisms, which are the structure-preserving functions from one ring to another. Next, we study ideals and quotient rings, which provide the most general version of modular arithmetic in a ring, and which are fundamentally connected with ring homomorphisms. We close with a detailed study of the structure of ideals and quotients in commutative rings with 1. 3.1 Ring Isomorphisms and Homomorphisms • We begin our study with a discussion of structure-preserving maps between rings. 3.1.1 Ring Isomorphisms • We have encountered several examples of rings with very similar structures. • For example, consider the two rings R = Z=6Z and S = (Z=2Z) × (Z=3Z).
    [Show full text]
  • Sets and Classes As Many
    SETS AND CLASSES AS MANY by John L. Bell INTRODUCTION Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual, as indicated in the diagram: Individuals Sets Classes If on the other hand we insist—as we shall here—that classes are to be taken in the sense of multitudes, pluralities, or classes as many, then no class can be an individual and so, in particular, the concept of set will need to be redefined. Here by “class as many” we have in mind what Erik Stenius refers to in [5] as set of, which he defines as follows: If we start from a Universe of Discourse given in advance, then we may define a set-of things as being many things in this UoD or just one thing - or even no things, if we want to introduce this way of speaking. Stenius draws a sharp distinction between this concept and that of 2 set as a thing. As he says, The distinction between a set-as-a-thing and a set of corresponds to the Russellian distinction between a “class as one” and a “class as many” (Principles of Mathematics, p. 76). Only I use the expressions ‘set-as-a- thing” and “set-of” instead, in order to stress the (grammatical and) ontological character of the distinction; and also because of the difficulty that a set-of need not consist of many things; it can comprise just one thing or no things.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]
  • Ordinal Numbers and the Well-Ordering Theorem Ken Brown, Cornell University, September 2013
    Mathematics 4530 Ordinal numbers and the well-ordering theorem Ken Brown, Cornell University, September 2013 The ordinal numbers form an extension of the natural numbers. Here are the first few of them: 0; 1; 2; : : : ; !; ! + 1;! + 2;:::;! + ! =: !2;!2 + 1;:::;!3;:::;!2;:::: They go on forever. As soon as some initial segment of ordinals has been con- structed, a new one is adjoined that is bigger than all of them. The theory of ordinals is closely related to the theory of well-ordered sets (see Section 10 of Munkres). Recall that a simply ordered set X is said to be well- ordered if every nonempty subset Y has a smallest element, i.e., an element y0 such that y0 ≤ y for all y 2 Y . For example, the following sets are well-ordered: ;; f1g ; f1; 2g ; f1; 2; : : : ; ng ; f1; 2;::: g ; f1; 2;:::;!g ; f1; 2; : : : ; !; ! + 1g : On the other hand, Z is not well-ordered in its natural ordering. As the list of examples suggests, constructing arbitrarily large well-ordered sets is essentially the same as constructing the system of ordinal numbers. Rather than taking the time to develop the theory of ordinals, we will concentrate on well-ordered sets in what follows. Notation. In dealing with ordered sets in what follows we will often use notation such as X<x := fy 2 X j y < xg : A set of the form X<x is called a section of X. Well-orderings are useful because they allow proofs by induction: Induction principle. Let X be a well-ordered set and let Y be a subset.
    [Show full text]
  • Cofinality of the Nonstationary Ideal 0
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 12, Pages 4813–4837 S 0002-9947(05)04007-9 Article electronically published on June 29, 2005 COFINALITY OF THE NONSTATIONARY IDEAL PIERRE MATET, ANDRZEJ ROSLANOWSKI, AND SAHARON SHELAH Abstract. We show that the reduced cofinality of the nonstationary ideal NSκ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NSκ is the least cardinality of any family F of nonsta- tionary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F. For this we investigate connections of κ the various cofinalities of NSκ with other cardinal characteristics of κ and we also give a property of forcing notions (called manageability) which is pre- served in <κ–support iterations and which implies that the forcing notion preserves non-meagerness of subsets of κκ (and does not collapse cardinals nor changes cofinalities). 0. Introduction Let κ be a regular uncountable cardinal. For C ⊆ κ and γ ≤ κ,wesaythatγ is a limit point of C if (C ∩ γ)=γ>0. C is closed unbounded if C is a cofinal subset of κ containing all its limit points less than κ.AsetA ⊆ κ is nonstationary if A is disjoint from some closed unbounded subset C of κ. The nonstationary subsets of κ form an ideal on κ denoted by NSκ.Thecofinality of this ideal, cof(NSκ), is the least cardinality of a family F of nonstationary subsets of κ such that every nonstationary subset of κ is contained in a member of F.Thereduced cofinality of NSκ, cof(NSκ), is the least cardinality of a family F⊆NSκ such that every nonstationary subset of κ can be covered by less than κ many members of F.This paper addresses the question of whether cof(NSκ)=cof(NSκ).
    [Show full text]
  • Lecture 3: Constructing the Natural Numbers 1 Building
    Math/CS 120: Intro. to Math Professor: Padraic Bartlett Lecture 3: Constructing the Natural Numbers Weeks 3-4 UCSB 2014 When we defined what a proof was in our first set of lectures, we mentioned that we wanted our proofs to only start by assuming \true" statements, which we said were either previously proven-to-be-true statements or a small handful of axioms, mathematical statements which we are assuming to be true. At the time, we \handwaved" away what those axioms were, in favor of using known properties/definitions to prove results! In this talk, however, we're going to delve into the bedrock of exactly \what" properties are needed to build up some of our favorite number systems. 1 Building the Natural Numbers 1.1 First attempts. Intuitively, we think of the natural numbers as the following set: Definition. The natural numbers, denoted as N, is the set of the positive whole numbers. We denote it as follows: N = f0; 1; 2; 3;:::g This is a fine definition for most of the mathematics we will perform in this class! However, suppose that you were feeling particularly paranoid about your fellow mathematicians; i.e. you have a sneaking suspicion that the Goldbach conjecture is false, and it somehow boils down to the natural numbers being ill-defined. Or you think that you can prove P = NP with the corollary that P = NP, and to do that you need to figure out what people mean by these blackboard-bolded letters. Or you just wanted to troll your professors in CCS.
    [Show full text]
  • Preliminary Knowledge on Set Theory
    Huan Long Shanghai Jiao Tong University Basic set theory Relation Function Spring 2018 Georg Cantor(1845-1918) oGerman mathematician o Founder of set theory Bertrand Russell(1872-1970) oBritish philosopher, logician, mathematician, historian, and social critic. Ernst Zermelo(1871-1953) oGerman mathematician, foundations of mathematics and hence on philosophy David Hilbert (1862-1943) o German mathematicia, one of the most influential and universal mathematicians of the 19th and early 20th centuries. Kurt Gödel(1906-1978) oAustrian American logician, mathematician, and philosopher. ZFC not ¬CH . Paul Cohen(1934-2007)⊢ oAmerican mathematician, 1963: ZFC not CH,AC . Spring 2018 ⊢ Spring 2018 By Georg Cantor in 1870s: A set is an unordered collection of objects. ◦ The objects are called the elements, or members, of the set. A set is said to contain its elements. Notation: ◦ Meaning that: is an element of the set A, or, Set A contains ∈ . Important: ◦ Duplicates do not matter. ◦ Order does not matter. Spring 2018 a∈A a is an element of the set A. a A a is NOT an element of the set A. Set of sets {{a,b},{1, 5.2}, k} ∉ the empty set, or the null set, is set that has no elements. A B subset relation. Each element of A is also an element of B. ∅ A=B equal relation. A B and B A. ⊆ A B ⊆ ⊆ A B strict subset relation. If A B and A B ≠ |A| cardinality of a set, or the number of distinct elements. ⊂ ⊆ ≠ Venn Diagram V U A B U Spring 2018 { , , , } a {{a}} ∈ ∉ {{ }} ∅ ∉∅ {3,4,5}={5,4,3,4} ∅ ∈ ∅ ∈ ∅ S { } ∅⊆ S S ∅ ⊂ ∅ |{3, 3, 4, {2, 3},{1,2,{f}} }|=4 ⊆ Spring 2018 Union Intersection Difference Complement Symmetric difference Power set Spring 2018 Definition Let A and B be sets.
    [Show full text]
  • Elements of Set Theory
    Elements of Set Theory P´eterMekis Department of Logic, ELTE Budapest December 11, 2017 Preface This is an as yet incomplete draft of what is going to be the lecture notes for a course under the same title for students of the Logic and the Theory of Science MA program of ELTE Budapest. The text is written for students that are logically and philosophically minded, but do not necessarily have a mathematical background. However, it requires familiarity with the basics of first-order logic. A reader who is skilled in reading mathematical texts will find most of the proofs too detailed. Note that any proof can be skipped and taken as an exercise to avoid boredom. Instead of the more well-known system of Zermelo{Fraenkel set theory, these notes use the framework of G¨odel{Bernays set theory (GB). Beyond the personal taste of the author, the reason for this choice is that there is a follow-up of this course under the title Alternative set theories, the basic task of which is to investigate the philosophical motivations and implications of various versions of the set/class distinction, which is at the heart of any axiomatic set theory, and which is more convenient to discuss in a language where one can quantify over classes. A mathematically minded reader may be surprised by the length of discussing certain basic topics, even without any apparent practical profit, and even at the cost of giving a rather sketchy presentation of more advanced results. An example is the detailed com- parison of three different ways to represent natural numbers in GB.
    [Show full text]