DOCSLIB.ORG

Producing New Bijections From

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

ro duing xew fijetions from yld @to pp er in edvnes in wthemtisA epril PPD IWWR hvid peldmn niversity of xew rmpshire tmes ropp wsshusetts snstitute of ehnology efeg por ny purely omintoril onstrution tht pro dues new nite sets from given onesD ijetion etween two given sets nturlly determines ijetion etween the orresp onding onstruted setsF e investigte the p ossiility of going in reverseX dening ijetion etween the originl o jets from given ijetion etween the onstruted o jetsD in eet neling4 the onstrutionF e present preise formultion of this question nd then give onrete riterion for determining whether suh nelltion is p ossileF sf the riterion is stisedD then nelltion pro edure existsD nd indeedD for mny of the onstrutions tht we hve studied p olynomilEtime nellE tion pro edure hs een foundY on the other hndD when the riterion is not stisedD no pro edureD however omplitedD n p erform the nelltionF he elerted involution priniple of qrsi nd wilne ts into our ruri one it is reognized s thinly disguised form of neling disjoint union with xed setD itself well{known tehnique tht ts into our theoretil frmeworkF e show tht the onstrution tht forms the grtesin pro dut of given set with some otherD xed set nnot in generl e neledD ut tht it n e neled in the se where the xed set rries distinguished elementF e lso show tht the onstrution tht tkes the mth grtesin p ower of given set n e neledD ut tht the onstrution tht forms the p ower set of given set nnotF uey wordsX fijetionD ijetive pro ofD qEsetD grtesin p owerD p ower set IF sntro dution emong the vrious mens of estlishing omintoril identitiesD mny omE intorilists esp eilly prize ijetive pro ofs4X diret demonstrtions y mens of mnifest ijetionsF he present sttus of the siene of ijetive pro of my e likened to tht of rulerEndEompss onstrutions prior to quss nd ntzelF iven with mny ingenious exmplesD ijetive pro of 4 remins n informl notionF he lk of suitle formliztion prevents exE plortion of the limittions of ijetive pro ofD4 nd so no results we know of my e interpreted to syD in ny senseD tht there exist nlytilly verile omintoril identities dmitting no ijetive pro ofF sndeed the sitution my well e sensitive to how ijetive pro of 4 is mde preiseD muh s nE gles my e triseted with mrked strightedgesD ut not with unmrked strightedgesF sn this pp erD we show tht just s lgeri metho ds @s emo died in qlois theoryA n e rought to er on questions of onstrutiility of geometriE l o jets y use of geometril to olsD so to o n metho ds from lger e pplied to questions of onstrutiility of omintoril o jets vi omiE ntoril to ols suh s ijetionsF e do not oer sp ei formliztion of the notion of ijetive pro of 4 in this pp erF ther we oerD nd exploreD distintion etween wek ijeE tive pro ofs nd strong ijetive pro ofs tht should e relevnt in virtully ny prtiulr forml ontextF vet us now explin this distintionF e omiE ntoril identity onerns two indexed fmilies of nite setsD sy fe g nd i iPs ff g D nd sserts tht for ll iD je j a jf jF por our purp oses nothing will i iPs i i e lost if we sp ek s though i were xedD so herefter we drop the susriptF xowD let @A e n ritrry onstrution tht pro dues new nite sets from old suh tht the rdinlity of determines the rdinlity of for ll F hen mnifest ijetion etween e nd f lso proves tht jej a jf jY we onsider this wek ijetive proof tht jej a jf jD insmuh s the ft tht the rdinlity of determines the rdinlity of is not itself proved ijetivelyF e strong ijetive proof tht jej a jf j must exhiit mnifest ijetion etween the sets e nd f themselvesF es we shll seeD for some onstrutions @A it is lwys p ossile to dene ijetion f X e 3 f just in terms of ny given ijetion p X e 3 f D nd I not tking into ount of the sp eil nture of the elements of the sets e nd f F rovided tht the denition of f from p is within one9s forml mensD one is then gurnteed tht wek ijetive proof using @A determines strong ijetive proofF por our purp osesD to sy tht the denition of f from p is within one9s forml mens4 is to sy tht f n e omputed from p D in senrio tht we will mke preise for the ske of deniteness @though the theorems tht we prove re not overly sensitive to the preise detils of the mo delAF hen the nture of @A p ermits us to ompute f from p D we sy tht suh @A n e eetively neledF e give exmples of onstrutions @A tht n e eetively neledD long with others tht nnotF wore generllyD supp ose tht @Y A is onstrution tht tkes two rguE ments nd tht g is some sort of strutured setD notion tht we shll mke preise lterF upp ose ijetion p X @eY g A 3 @f Y g A is givenF e derive riterion for deiding whether ijetion f X e 3 f n e dened solely in terms of p nd the given struture on g F his riterion is lgeri in ntureD nd in sp ei pplitions it devolves into questions out p ermuttion representtions of nite groupsF hen the riterion is stisedD nelltion priniple is gurnteed to existD nd we my pro eed to seek it outF sndeedD we hve typilly found tht when the nelility riterion is stisedD nelltion pro edure exists whose runningEtime is ounded y p olynomil funtion of the sizes of the sets in questionF sf the riterion is not stisedD then there is no p oint in serhing for nelE ltion pro edureF roweverD even suh negtive results hve p ositive imp ort of ertin kind for the working omintorilistF por instneD knowing tht the p ower set onstrution nnot generlly e neled suggests the e f p ossiility of lo oking for interesting ijetions etween P nd P even in situtions where no interesting ijetion etween e nd f is villeF sn ny seD questions out nelilityD when run through the mill of our generlEpurp ose mhineryD give rise to interesting questions out p ermutE tion representtions of nite groupsD nd thus n e resoure for lgerists in serh of novel questions to p onderF yur metho ds ome from group theory nd re purely mthemtilD s opE p osed to metmthemtilF xevertheless it my e fruitful to interpret our P results s sttements out omintoris in vrious top osesD though we do not pursue thisF yur work n e seen s following in trdition initited y edrino qrsi nd tephen wilne R in their study of the involution priniple nd ontinued y rer ert ilf IP nd fsil qordon S in their work on the priniple of inlusion nd exlusionF elso of some relevne is work of vszlo vovsz nd o ert eppleson on nite sets with reltionl strutures @see I nd the referenes ited thereinAF heir work diers from ours in tht we regrd two nite strutures s eing equivlent if they re denle from one notherD wheres those uthors require tht the strutures tully e isomorphiF xeverthelessD there re mny p oints of similrity etween their pp ers nd our pp erD in terms of oth metho ds used nd results otinedD nd it would e go o d to understnd etter the reltionship etween the two theoriesF yur pp er is orgnized s followsF sn etion P we give preliminries onE erning group tions nd x some nottionF xextD in etion Q we intro due proto ol to mke denite wht we men y eetive nelltionF hen we prove heorem ID the fsi iquivrine griterionD to formlize the intuE itive notion eetively nelle omintoril onstrution4 @etion RAD nd we give severl vritions thereof @etion SAF heorem P rests the fsi iquivrine griterion in more useful nd more elegnt form @eE tion TAF etion U digresses to tret some issues relted to vrint form of nelltionD for the ske of ompletenessF etion V shows tht issues of uniformity4 of nelltion pro eduresD whih re not ddressed elsewhere in the rtileD n e given preise formultion if one mo dies the frmeE work so tht the role plyed y tions of groups is tken over y tions of monoidsF he ve setions tht follow pply heorem P to otin onE rete resultsD some p ositive nd some negtiveD some new nd some oldX p ositive result for nelltion of disjoint union with xed set @oldD ut we oserve how it enompsses the elerted involution priniple of qrsi nd wilneD therey tting tht tehnique into our ruriAY negtive result for grtesin pro duts @oldD though not previously stted in this form in the litertureD t lest to our knowledgeAY p ositive result for grtesin pro dE uts with p ointed sets @newAY p ositive result for grtesin p owers @newAY nd generlly negtive result for p ower sets @newA whih is lried y the Q solution of n interesting groupEtheoretil prolemF etion IR gives slik pro of of very generl equivrine riterionF etion IS situtes our work in more strtD tegoryEtheoreti settingF pinllyD etion IT trets the relevne of our pp er to issues in the foundtions of mthemtisF yp en prolems re intersp ersed throughout the pp erF st is not neessry to understnd the foundtionl mteril in the rst hlf of the pp er to p eruse the vrious expliit exmples in etions W through IQY indeedD these setions ontin the min fruits of our workD nd the reder my wish to smple them efore studying the ro otsD trunkD nd rnhesF es middle ourseD the reder might strt with etions PD QD RD nd T efore pro eeding to etions W through IQF ome of our susidiry prop ositions hve trivil pro ofs tht my not e trivil for the reder to nd unidedY we hve inluded ompt versions of these pro ofsF roweverD rther thn try to follow these veritionsD the reder might nd it more helpful to drw the pproprite digrms nd hse rrows in the usul wyD onsulting our pro ofs only s lst resortF PF reliminries e denote the integers y D the nturl numers @inluding HA y xF ell other sets onsidered re ssumed niteF e qEset is set with n tion y group q @on the leftAF @por generl kground on group tions on nite setsD see ghpter I of TF sf r is sugroup of qD the osetEspe qar is lso qEsetD under the denition H H g
Recommended publications
  • Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of Abbreviations

    Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of Abbreviations

    Notes: c F.P. Greenleaf, 2000-2014 v43-s14sets.tex (version 1/1/2014) Algebra I Chapter 1. Basic Facts from Set Theory 1.1 Glossary of abbreviations. Below we list some standard math symbols that will be used as shorthand abbreviations throughout this course. means “for all; for every” • ∀ means “there exists (at least one)” • ∃ ! means “there exists exactly one” • ∃ s.t. means “such that” • = means “implies” • ⇒ means “if and only if” • ⇐⇒ x A means “the point x belongs to a set A;” x / A means “x is not in A” • ∈ ∈ N denotes the set of natural numbers (counting numbers) 1, 2, 3, • · · · Z denotes the set of all integers (positive, negative or zero) • Q denotes the set of rational numbers • R denotes the set of real numbers • C denotes the set of complex numbers • x A : P (x) If A is a set, this denotes the subset of elements x in A such that •statement { ∈ P (x)} is true. As examples of the last notation for specifying subsets: x R : x2 +1 2 = ( , 1] [1, ) { ∈ ≥ } −∞ − ∪ ∞ x R : x2 +1=0 = { ∈ } ∅ z C : z2 +1=0 = +i, i where i = √ 1 { ∈ } { − } − 1.2 Basic facts from set theory. Next we review the basic definitions and notations of set theory, which will be used throughout our discussions of algebra. denotes the empty set, the set with nothing in it • ∅ x A means that the point x belongs to a set A, or that x is an element of A. • ∈ A B means A is a subset of B – i.e.
  • MATH 361 Homework 9

    MATH 361 Homework 9

    MATH 361 Homework 9 Royden 3.3.9 First, we show that for any subset E of the real numbers, Ec + y = (E + y)c (translating the complement is equivalent to the complement of the translated set). Without loss of generality, assume E can be written as c an open interval (e1; e2), so that E + y is represented by the set fxjx 2 (−∞; e1 + y) [ (e2 + y; +1)g. This c is equal to the set fxjx2 = (e1 + y; e2 + y)g, which is equivalent to the set (E + y) . Second, Let B = A − y. From Homework 8, we know that outer measure is invariant under translations. Using this along with the fact that E is measurable: m∗(A) = m∗(B) = m∗(B \ E) + m∗(B \ Ec) = m∗((B \ E) + y) + m∗((B \ Ec) + y) = m∗(((A − y) \ E) + y) + m∗(((A − y) \ Ec) + y) = m∗(A \ (E + y)) + m∗(A \ (Ec + y)) = m∗(A \ (E + y)) + m∗(A \ (E + y)c) The last line follows from Ec + y = (E + y)c. Royden 3.3.10 First, since E1;E2 2 M and M is a σ-algebra, E1 [ E2;E1 \ E2 2 M. By the measurability of E1 and E2: ∗ ∗ ∗ c m (E1) = m (E1 \ E2) + m (E1 \ E2) ∗ ∗ ∗ c m (E2) = m (E2 \ E1) + m (E2 \ E1) ∗ ∗ ∗ ∗ c ∗ c m (E1) + m (E2) = 2m (E1 \ E2) + m (E1 \ E2) + m (E1 \ E2) ∗ ∗ ∗ c ∗ c = m (E1 \ E2) + [m (E1 \ E2) + m (E1 \ E2) + m (E1 \ E2)] c c Second, E1 \ E2, E1 \ E2, and E1 \ E2 are disjoint sets whose union is equal to E1 [ E2.
  • Fast and Private Computation of Cardinality of Set Intersection and Union

    Fast and Private Computation of Cardinality of Set Intersection and Union

    Fast and Private Computation of Cardinality of Set Intersection and Union Emiliano De Cristofaroy, Paolo Gastiz, and Gene Tsudikz yPARC zUC Irvine Abstract. With massive amounts of electronic information stored, trans- ferred, and shared every day, legitimate needs for sensitive information must be reconciled with natural privacy concerns. This motivates var- ious cryptographic techniques for privacy-preserving information shar- ing, such as Private Set Intersection (PSI) and Private Set Union (PSU). Such techniques involve two parties { client and server { each with a private input set. At the end, client learns the intersection (or union) of the two respective sets, while server learns nothing. However, if desired functionality is private computation of cardinality rather than contents, of set intersection, PSI and PSU protocols are not appropriate, as they yield too much information. In this paper, we design an efficient crypto- graphic protocol for Private Set Intersection Cardinality (PSI-CA) that yields only the size of set intersection, with security in the presence of both semi-honest and malicious adversaries. To the best of our knowl- edge, it is the first protocol that achieves complexities linear in the size of input sets. We then show how the same protocol can be used to pri- vately compute set union cardinality. We also design an extension that supports authorization of client input. 1 Introduction Proliferation of, and growing reliance on, electronic information trigger the increase in the amount of sensitive data stored and processed in cyberspace. Consequently, there is a strong need for efficient cryptographic techniques that allow sharing information with privacy. Among these, Private Set Intersection (PSI) [11, 24, 14, 21, 15, 22, 9, 8, 18], and Private Set Union (PSU) [24, 15, 17, 12, 32] have attracted a lot of attention from the research community.
  • 1 Measurable Sets

    1 Measurable Sets

    Math 4351, Fall 2018 Chapter 11 in Goldberg 1 Measurable Sets Our goal is to define what is meant by a measurable set E ⊆ [a; b] ⊂ R and a measurable function f :[a; b] ! R. We defined the length of an open set and a closed set, denoted as jGj and jF j, e.g., j[a; b]j = b − a: We will use another notation for complement and the notation in the Goldberg text. Let Ec = [a; b] n E = [a; b] − E. Also, E1 n E2 = E1 − E2: Definitions: Let E ⊆ [a; b]. Outer measure of a set E: m(E) = inffjGj : for all G open and E ⊆ Gg. Inner measure of a set E: m(E) = supfjF j : for all F closed and F ⊆ Eg: 0 ≤ m(E) ≤ m(E). A set E is a measurable set if m(E) = m(E) and the measure of E is denoted as m(E). The symmetric difference of two sets E1 and E2 is defined as E1∆E2 = (E1 − E2) [ (E2 − E1): A set is called an Fσ set (F-sigma set) if it is a union of a countable number of closed sets. A set is called a Gδ set (G-delta set) if it is a countable intersection of open sets. Properties of Measurable Sets on [a; b]: 1. If E1 and E2 are subsets of [a; b] and E1 ⊆ E2, then m(E1) ≤ m(E2) and m(E1) ≤ m(E2). In addition, if E1 and E2 are measurable subsets of [a; b] and E1 ⊆ E2, then m(E1) ≤ m(E2).
  • Solutions to Homework 2 Math 55B

    Solutions to Homework 2 Math 55B

    Solutions to Homework 2 Math 55b 1. Give an example of a subset A ⊂ R such that, by repeatedly taking closures and complements. Take A := (−5; −4)−Q [[−4; −3][f2g[[−1; 0][(1; 2)[(2; 3)[ (4; 5)\Q . (The number of intervals in this sets is unnecessarily large, but, as Kevin suggested, taking a set that is complement-symmetric about 0 cuts down the verification in half. Note that this set is the union of f0g[(1; 2)[(2; 3)[ ([4; 5] \ Q) and the complement of the reflection of this set about the the origin). Because of this symmetry, we only need to verify that the closure- complement-closure sequence of the set f0g [ (1; 2) [ (2; 3) [ (4; 5) \ Q consists of seven distinct sets. Here are the (first) seven members of this sequence; from the eighth member the sets start repeating period- ically: f0g [ (1; 2) [ (2; 3) [ (4; 5) \ Q ; f0g [ [1; 3] [ [4; 5]; (−∞; 0) [ (0; 1) [ (3; 4) [ (5; 1); (−∞; 1] [ [3; 4] [ [5; 1); (1; 3) [ (4; 5); [1; 3] [ [4; 5]; (−∞; 1) [ (3; 4) [ (5; 1). Thus, by symmetry, both the closure- complement-closure and complement-closure-complement sequences of A consist of seven distinct sets, and hence there is a total of 14 different sets obtained from A by taking closures and complements. Remark. That 14 is the maximal possible number of sets obtainable for any metric space is the Kuratowski complement-closure problem. This is proved by noting that, denoting respectively the closure and com- plement operators by a and b and by e the identity operator, the relations a2 = a; b2 = e, and aba = abababa take place, and then one can simply list all 14 elements of the monoid ha; b j a2 = a; b2 = e; aba = abababai.
  • ON the CONSTRUCTION of NEW TOPOLOGICAL SPACES from EXISTING ONES Consider a Function F

    ON the CONSTRUCTION of NEW TOPOLOGICAL SPACES from EXISTING ONES Consider a Function F

    ON THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM EXISTING ONES EMILY RIEHL Abstract. In this note, we introduce a guiding principle to define topologies for a wide variety of spaces built from existing topological spaces. The topolo- gies so-constructed will have a universal property taking one of two forms. If the topology is the coarsest so that a certain condition holds, we will give an elementary characterization of all continuous functions taking values in this new space. Alternatively, if the topology is the finest so that a certain condi- tion holds, we will characterize all continuous functions whose domain is the new space. Consider a function f : X ! Y between a pair of sets. If Y is a topological space, we could define a topology on X by asking that it is the coarsest topology so that f is continuous. (The finest topology making f continuous is the discrete topology.) Explicitly, a subbasis of open sets of X is given by the preimages of open sets of Y . With this definition, a function W ! X, where W is some other space, is continuous if and only if the composite function W ! Y is continuous. On the other hand, if X is assumed to be a topological space, we could define a topology on Y by asking that it is the finest topology so that f is continuous. (The coarsest topology making f continuous is the indiscrete topology.) Explicitly, a subset of Y is open if and only if its preimage in X is open. With this definition, a function Y ! Z, where Z is some other space, is continuous if and only if the composite function X ! Z is continuous.
  • MAD FAMILIES CONSTRUCTED from PERFECT ALMOST DISJOINT FAMILIES §1. Introduction. a Family a of Infinite Subsets of Ω Is Called

    MAD FAMILIES CONSTRUCTED from PERFECT ALMOST DISJOINT FAMILIES §1. Introduction. a Family a of Infinite Subsets of Ω Is Called

    The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000 MAD FAMILIES CONSTRUCTED FROM PERFECT ALMOST DISJOINT FAMILIES JORG¨ BRENDLE AND YURII KHOMSKII 1 Abstract. We prove the consistency of b > @1 together with the existence of a Π1- definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Ques- tion 16]. For the proof we construct a mad family in L which is an @1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number aB , defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of aB < b (and hence, aB < a). x1. Introduction. A family A of infinite subsets of ! is called almost disjoint (a.d.) if any two elements a; b of A have finite intersection. A family A is called maximal almost disjoint, or mad, if it is an infinite a.d. family which is maximal with respect to that property|in other words, 8a 9b 2 A (ja \ bj = !). The starting point of this paper is the following theorem of Adrian Mathias [11, Corollary 4.7]: Theorem 1.1 (Mathias). There are no analytic mad families. 1 On the other hand, it is easy to see that in L there is a Σ2 definable mad family. In [12, Theorem 8.23], Arnold Miller used a sophisticated method to 1 prove the seemingly stronger result that in L there is a Π1 definable mad family.
  • Julia's Efficient Algorithm for Subtyping Unions and Covariant

    Julia's Efficient Algorithm for Subtyping Unions and Covariant

    Julia’s Efficient Algorithm for Subtyping Unions and Covariant Tuples Benjamin Chung Northeastern University, Boston, MA, USA [email protected] Francesco Zappa Nardelli Inria of Paris, Paris, France [email protected] Jan Vitek Northeastern University, Boston, MA, USA Czech Technical University in Prague, Czech Republic [email protected] Abstract The Julia programming language supports multiple dispatch and provides a rich type annotation language to specify method applicability. When multiple methods are applicable for a given call, Julia relies on subtyping between method signatures to pick the correct method to invoke. Julia’s subtyping algorithm is surprisingly complex, and determining whether it is correct remains an open question. In this paper, we focus on one piece of this problem: the interaction between union types and covariant tuples. Previous work normalized unions inside tuples to disjunctive normal form. However, this strategy has two drawbacks: complex type signatures induce space explosion, and interference between normalization and other features of Julia’s type system. In this paper, we describe the algorithm that Julia uses to compute subtyping between tuples and unions – an algorithm that is immune to space explosion and plays well with other features of the language. We prove this algorithm correct and complete against a semantic-subtyping denotational model in Coq. 2012 ACM Subject Classification Theory of computation → Type theory Keywords and phrases Type systems, Subtyping, Union types Digital Object Identifier 10.4230/LIPIcs.ECOOP.2019.24 Category Pearl Supplement Material ECOOP 2019 Artifact Evaluation approved artifact available at https://dx.doi.org/10.4230/DARTS.5.2.8 Acknowledgements The authors thank Jiahao Chen for starting us down the path of understanding Julia, and Jeff Bezanson for coming up with Julia’s subtyping algorithm.
  • Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations on R with All Classes Countable Is the Main Result of the Paper

    Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations on R with All Classes Countable Is the Main Result of the Paper

    CARDINALITY OF WELLORDERED DISJOINT UNIONS OF QUOTIENTS OF SMOOTH EQUIVALENCE RELATIONS WILLIAM CHAN AND STEPHEN JACKSON Abstract. Assume ZF + AD+ + V = L(P(R)). Let ≈ denote the relation of being in bijection. Let κ ∈ ON and hEα : α<κi be a sequence of equivalence relations on R with all classes countable and for all α<κ, R R R R R /Eα ≈ . Then the disjoint union Fα<κ /Eα is in bijection with × κ and Fα<κ /Eα has the J´onsson property. + <ω Assume ZF + AD + V = L(P(R)). A set X ⊆ [ω1] 1 has a sequence hEα : α<ω1i of equivalence relations on R such that R/Eα ≈ R and X ≈ F R/Eα if and only if R ⊔ ω1 injects into X. α<ω1 ω ω Assume AD. Suppose R ⊆ [ω1] × R is a relation such that for all f ∈ [ω1] , Rf = {x ∈ R : R(f,x)} ω is nonempty and countable. Then there is an uncountable X ⊆ ω1 and function Φ : [X] → R which uniformizes R on [X]ω: that is, for all f ∈ [X]ω, R(f, Φ(f)). Under AD, if κ is an ordinal and hEα : α<κi is a sequence of equivalence relations on R with all classes ω R countable, then [ω1] does not inject into Fα<κ /Eα. 1. Introduction The original motivation for this work comes from the study of a simple combinatorial property of sets using only definable methods. The combinatorial property of concern is the J´onsson property: Let X be any n n <ω n set.
  • Disjoint Union / Find Equivalence Relations Equivalence Classes

    Disjoint Union / Find Equivalence Relations Equivalence Classes

    Equivalence Relations Disjoint Union / Find • A relation R is defined on set S if for CSE 326 every pair of elements a, b∈S, a R b is Data Structures either true or false. Unit 13 • An equivalence relation is a relation R that satisfies the 3 properties: › Reflexive: a R a for all a∈S Reading: Chapter 8 › Symmetric: a R b iff b R a; for all a,b∈S › Transitive: a R b and b R c implies a R c 2 Dynamic Equivalence Equivalence Classes Problem • Given an equivalence relation R, decide • Starting with each element in a singleton set, whether a pair of elements a,b∈S is and an equivalence relation, build the equivalence classes such that a R b. • Requires two operations: • The equivalence class of an element a › Find the equivalence class (set) of a given is the subset of S of all elements element › Union of two sets related to a. • It is a dynamic (on-line) problem because the • Equivalence classes are disjoint sets sets change during the operations and Find must be able to cope! 3 4 Disjoint Union - Find Union • Maintain a set of disjoint sets. • Union(x,y) – take the union of two sets › {3,5,7} , {4,2,8}, {9}, {1,6} named x and y • Each set has a unique name, one of its › {3,5,7} , {4,2,8}, {9}, {1,6} members › Union(5,1) › {3,5,7} , {4,2,8}, {9}, {1,6} {3,5,7,1,6}, {4,2,8}, {9}, 5 6 Find An Application • Find(x) – return the name of the set • Build a random maze by erasing edges.
  • Handout #4: Connected Metric Spaces

    Handout #4: Connected Metric Spaces

    Connected Sets Math 331, Handout #4 You probably have some intuitive idea of what it means for a metric space to be \connected." For example, the real number line, R, seems to be connected, but if you remove a point from it, it becomes \disconnected." However, it is not really clear how to define connected metric spaces in general. Furthermore, we want to say what it means for a subset of a metric space to be connected, and that will require us to look more closely at subsets of metric spaces than we have so far. We start with a definition of connected metric space. Note that, similarly to compactness and continuity, connectedness is actually a topological property rather than a metric property, since it can be defined entirely in terms of open sets. Definition 1. Let (M; d) be a metric space. We say that (M; d) is a connected metric space if and only if M cannot be written as a disjoint union N = X [ Y where X and Y are both non-empty open subsets of M. (\Disjoint union" means that M = X [ Y and X \ Y = ?.) A metric space that is not connected is said to be disconnnected. Theorem 1. A metric space (M; d) is connected if and only if the only subsets of M that are both open and closed are M and ?. Equivalently, (M; d) is disconnected if and only if it has a non-empty, proper subset that is both open and closed. Proof. Suppose (M; d) is a connected metric space.
  • Naïve Set Theory Basic Definitions Naïve Set Theory Is the Non-Axiomatic Treatment of Set Theory

    Naïve Set Theory Basic Definitions Naïve Set Theory Is the Non-Axiomatic Treatment of Set Theory

    Naïve Set Theory Basic Definitions Naïve set theory is the non-axiomatic treatment of set theory. In the axiomatic treatment, which we will only allude to at times, a set is an undefined term. For us however, a set will be thought of as a collection of some (possibly none) objects. These objects are called the members (or elements) of the set. We use the symbol "∈" to indicate membership in a set. Thus, if A is a set and x is one of its members, we write x ∈ A and say "x is an element of A" or "x is in A" or "x is a member of A". Note that "∈" is not the same as the Greek letter "ε" epsilon. Basic Definitions Sets can be described notationally in many ways, but always using the set brackets "{" and "}". If possible, one can just list the elements of the set: A = {1,3, oranges, lions, an old wad of gum} or give an indication of the elements: ℕ = {1,2,3, ... } ℤ = {..., -2,-1,0,1,2, ...} or (most frequently in mathematics) using set-builder notation: S = {x ∈ ℝ | 1 < x ≤ 7 } or {x ∈ ℝ : 1 < x ≤ 7 } which is read as "S is the set of real numbers x, such that x is greater than 1 and less than or equal to 7". In some areas of mathematics sets may be denoted by special notations. For instance, in analysis S would be written (1,7]. Basic Definitions Note that sets do not contain repeated elements. An element is either in or not in a set, never "in the set 5 times" for instance.