CSE 373: Data Structures & Algorithms Disjoint Sets & Union
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A Class of Symmetric Difference-Closed Sets Related to Commuting Involutions John Campbell
A class of symmetric difference-closed sets related to commuting involutions John Campbell To cite this version: John Campbell. A class of symmetric difference-closed sets related to commuting involutions. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1. hal-01345066v4 HAL Id: hal-01345066 https://hal.archives-ouvertes.fr/hal-01345066v4 Submitted on 18 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 19:1, 2017, #8 A class of symmetric difference-closed sets related to commuting involutions John M. Campbell York University, Canada received 19th July 2016, revised 15th Dec. 2016, 1st Feb. 2017, accepted 10th Feb. 2017. Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in Sn and An. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in Sn and An. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint 2-subsets of [n], and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in An. -
Natural Numerosities of Sets of Tuples
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 1, January 2015, Pages 275–292 S 0002-9947(2014)06136-9 Article electronically published on July 2, 2014 NATURAL NUMEROSITIES OF SETS OF TUPLES MARCO FORTI AND GIUSEPPE MORANA ROCCASALVO Abstract. We consider a notion of “numerosity” for sets of tuples of natural numbers that satisfies the five common notions of Euclid’s Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a no- tion, we show that, contrasting to cardinal arithmetic, the natural “Cantorian” definitions of order relation and arithmetical operations provide a very good algebraic structure. In fact, numerosities can be taken as the non-negative part of a discretely ordered ring, namely the quotient of a formal power series ring modulo a suitable (“gauge”) ideal. In particular, special numerosities, called “natural”, can be identified with the semiring of hypernatural numbers of appropriate ultrapowers of N. Introduction In this paper we consider a notion of “equinumerosity” on sets of tuples of natural numbers, i.e. an equivalence relation, finer than equicardinality, that satisfies the celebrated five Euclidean common notions about magnitudines ([8]), including the principle that “the whole is greater than the part”. This notion preserves the basic properties of equipotency for finite sets. In particular, the natural Cantorian definitions endow the corresponding numerosities with a structure of a discretely ordered semiring, where 0 is the size of the emptyset, 1 is the size of every singleton, and greater numerosity corresponds to supersets. This idea of “Euclidean” numerosity has been recently investigated by V. -
MTH 304: General Topology Semester 2, 2017-2018
MTH 304: General Topology Semester 2, 2017-2018 Dr. Prahlad Vaidyanathan Contents I. Continuous Functions3 1. First Definitions................................3 2. Open Sets...................................4 3. Continuity by Open Sets...........................6 II. Topological Spaces8 1. Definition and Examples...........................8 2. Metric Spaces................................. 11 3. Basis for a topology.............................. 16 4. The Product Topology on X × Y ...................... 18 Q 5. The Product Topology on Xα ....................... 20 6. Closed Sets.................................. 22 7. Continuous Functions............................. 27 8. The Quotient Topology............................ 30 III.Properties of Topological Spaces 36 1. The Hausdorff property............................ 36 2. Connectedness................................. 37 3. Path Connectedness............................. 41 4. Local Connectedness............................. 44 5. Compactness................................. 46 6. Compact Subsets of Rn ............................ 50 7. Continuous Functions on Compact Sets................... 52 8. Compactness in Metric Spaces........................ 56 9. Local Compactness.............................. 59 IV.Separation Axioms 62 1. Regular Spaces................................ 62 2. Normal Spaces................................ 64 3. Tietze's extension Theorem......................... 67 4. Urysohn Metrization Theorem........................ 71 5. Imbedding of Manifolds.......................... -
Old Notes from Warwick, Part 1
Measure Theory, MA 359 Handout 1 Valeriy Slastikov Autumn, 2005 1 Measure theory 1.1 General construction of Lebesgue measure In this section we will do the general construction of σ-additive complete measure by extending initial σ-additive measure on a semi-ring to a measure on σ-algebra generated by this semi-ring and then completing this measure by adding to the σ-algebra all the null sets. This section provides you with the essentials of the construction and make some parallels with the construction on the plane. Throughout these section we will deal with some collection of sets whose elements are subsets of some fixed abstract set X. It is not necessary to assume any topology on X but for simplicity you may imagine X = Rn. We start with some important definitions: Definition 1.1 A nonempty collection of sets S is a semi-ring if 1. Empty set ? 2 S; 2. If A 2 S; B 2 S then A \ B 2 S; n 3. If A 2 S; A ⊃ A1 2 S then A = [k=1Ak, where Ak 2 S for all 1 ≤ k ≤ n and Ak are disjoint sets. If the set X 2 S then S is called semi-algebra, the set X is called a unit of the collection of sets S. Example 1.1 The collection S of intervals [a; b) for all a; b 2 R form a semi-ring since 1. empty set ? = [a; a) 2 S; 2. if A 2 S and B 2 S then A = [a; b) and B = [c; d). -
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. -
Set Theory in Computer Science a Gentle Introduction to Mathematical Modeling I
Set Theory in Computer Science A Gentle Introduction to Mathematical Modeling I Jose´ Meseguer University of Illinois at Urbana-Champaign Urbana, IL 61801, USA c Jose´ Meseguer, 2008–2010; all rights reserved. February 28, 2011 2 Contents 1 Motivation 7 2 Set Theory as an Axiomatic Theory 11 3 The Empty Set, Extensionality, and Separation 15 3.1 The Empty Set . 15 3.2 Extensionality . 15 3.3 The Failed Attempt of Comprehension . 16 3.4 Separation . 17 4 Pairing, Unions, Powersets, and Infinity 19 4.1 Pairing . 19 4.2 Unions . 21 4.3 Powersets . 24 4.4 Infinity . 26 5 Case Study: A Computable Model of Hereditarily Finite Sets 29 5.1 HF-Sets in Maude . 30 5.2 Terms, Equations, and Term Rewriting . 33 5.3 Confluence, Termination, and Sufficient Completeness . 36 5.4 A Computable Model of HF-Sets . 39 5.5 HF-Sets as a Universe for Finitary Mathematics . 43 5.6 HF-Sets with Atoms . 47 6 Relations, Functions, and Function Sets 51 6.1 Relations and Functions . 51 6.2 Formula, Assignment, and Lambda Notations . 52 6.3 Images . 54 6.4 Composing Relations and Functions . 56 6.5 Abstract Products and Disjoint Unions . 59 6.6 Relating Function Sets . 62 7 Simple and Primitive Recursion, and the Peano Axioms 65 7.1 Simple Recursion . 65 7.2 Primitive Recursion . 67 7.3 The Peano Axioms . 69 8 Case Study: The Peano Language 71 9 Binary Relations on a Set 73 9.1 Directed and Undirected Graphs . 73 9.2 Transition Systems and Automata . -
Measures 1 Introduction
Measures These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland. 1 Introduction Our motivation for studying measure theory is to lay a foundation for mod- eling probabilities. I want to give a bit of motivation for the structure of measures that has developed by providing a sort of history of thought of measurement. Not really history of thought, this is more like a fictionaliza- tion (i.e. with the story but minus the proofs of assertions) of a standard treatment of Lebesgue measure that you can find, for example, in Capin- ski and Kopp, Chapter 2, or in Royden, books that begin by first studying Lebesgue measure on <. For studying probability, we have to study measures more generally; that will follow the introduction. People were interested in measuring length, area, volume etc. Let's stick to length. How was one to extend the notion of the length of an interval (a; b), l(a; b) = b − a to more general subsets of <? Given an interval I of any type (closed, open, left-open-right-closed, etc.), let l(I) be its length (the difference between its larger and smaller endpoints). Then the notion of Lebesgue outer measure (LOM) of a set A 2 < was defined as follows. Cover A with a countable collection of intervals, and measure the sum of lengths of these intervals. Take the smallest such sum, over all countable collections of ∗ intervals that cover A, to be the LOM of A. That is, m (A) = inf ZA, where 1 1 X 1 ZA = f l(In): A ⊆ [n=1Ing n=1 ∗ (the In referring to intervals). -
Disjoint Sets and Minimum Spanning Trees Introduction Disjoint-Set Data
COMS21103 Introduction I In this lecture we will start by discussing a data structure used for maintaining disjoint subsets of some bigger set. Disjoint sets and minimum spanning trees I This has a number of applications, including to maintaining connected Ashley Montanaro components of a graph, and to finding minimum spanning trees in [email protected] undirected graphs. Department of Computer Science, University of Bristol I We will then discuss two algorithms for finding minimum spanning Bristol, UK trees: an algorithm by Kruskal based on disjoint-set structures, and an algorithm by Prim which is similar to Dijkstra’s algorithm. 11 November 2013 I In both cases, we will see that efficient implementations of data structures give us efficient algorithms. Ashley Montanaro Ashley Montanaro [email protected] [email protected] COMS21103: Disjoint sets and MSTs Slide 1/48 COMS21103: Disjoint sets and MSTs Slide 2/48 Disjoint-set data structure Example A disjoint-set data structure maintains a collection = S ,..., S of S { 1 k } disjoint subsets of some larger “universe” U. Operation Returns S The data structure supports the following operations: (start) (empty) MakeSet(a) a { } 1. MakeSet(x): create a new set whose only member is x. As the sets MakeSet(b) a , b { } { } are disjoint, we require that x is not contained in any of the other sets. FindSet(b) b a , b { } { } 2. Union(x, y): combine the sets containing x and y (call these S , S ) to Union(a, b) a, b x y { } replace them with a new set S S . -
1 Basics About Sets
Math 20, Probability Spring 2018 S. Nanda SET THEORY 1 BASICS ABOUT SETS Probability theory uses the language of sets. As we have seen probability is defined and calculated for sets. This is a review of some basic concepts from set theory that are used in this class. Definition: A set is a collection of some items (elements). We often use capital letters to denote a set. To define a set we can list all the elements, or describe what the set contains in curly brackets. For example • a set A consists of the two elements | and ~. We write A = f|; ~g: • a set of positive even integers can be written thus: A = f2; 4; 6; 8;:::g where the ::: stand for all the subsequent even integers. We could also write A = f2k j k 2 Z+g. We always use curly brackets to denote the collection of elements in a set. To say that a belongs to A, we write a 2 A. To say that an element does not belong to a set, we use 2=. For example, we may write 1 2= A if A is the set of even integers. Note that ordering does not matter, so the two sets f|; ~g and f~; |g are equal. Some important sets used in math are given below. • The set of natural numbers, N = f1; 2; 3;:::g: • The set of integers, Z = f:::; −3; −2; −1; 0; 1; 2; 3;:::g: • The set of rational numbers Q. • The set of real numbers R: We can also define a set by mathematically stating the properties satisfied by the elements in the set. -
1. Σ-Algebras Definition 1. Let X Be Any Set and Let F Be a Collection Of
1. σ-algebras Definition 1. Let X be any set and let F be a collection of subsets of X. We say that F is a σ-algebra (on X), if it satisfies the following. (1) X 2 F. (2) If A 2 F, then Ac 2 F. 1 (3) If A1;A2; · · · 2 F, a countable collection, then [n=1An 2 F. In the above situation, the elements of F are called measurable sets and X (or more precisely (X; F)) is called a measurable space. Example 1. For any set X, P(X), the set of all subsets of X is a σ-algebra and so is F = f;;Xg. Theorem 1. Let X be any set and let G be any collection of subsets of X. Then there exists a σ-algebra, containing G and smallest with respect to inclusion. Proof. Let S be the collection of all σ-algebras containing G. This set is non-empty, since P(X) 2 S. Then F = \A2SA can easily be checked to have all the properties asserted in the theorem. Remark 1. In the above situation, F is called the σ-algebra generated by G. Definition 2. Let X be a topological space (for example, a metric space) and let B be the σ-algebra generated by the set of all open sets of X. The elements of B are called Borel sets and (X; B), a Borel measurable space. Definition 3. Let (X; F) be a measurable space and let Y be any topo- logical space and let f : X ! Y be any function. -
Disjoint Sets Andreas Klappenecker [Using Notes by R
Disjoint Sets Andreas Klappenecker [using notes by R. Sekar] Equivalence Classes Frequently, we declare an equivalence relation on a set S. So the elements of the set S are partitioned into equivalence classes. Two equivalence classes are either the same or disjoint. Example: S={1,2,3,4,5,6} Equivalence classes: {1}, {2,4}, {3,5,6}. The set S is partitioned into equivalence classes. Network connectivity Basic abstractions •Examplesset of objects • union command: connect two objects • find query: is there a path connecting one object to another? Is there a path from A to B? Connected components form a partition of the set of nodes. 4 Kruskal’s Minimum Spanning Tree Algorithm The vertices are partitioned into a forest of trees. Need: Efficient way to dynamically change the equivalence relation. Disjoint Sets How can we represent the elements of disjoint sets? Declare a representative element for each set. Implementation: Inverted trees. Each element points to parent. Root of the tree is the representative element. Intro Aggregate Charging Potential Table resizing Disjoint sets Inverted Trees Union by Depth Threaded Trees Path compression Disjoint Sets Represent disjoint sets as “inverted trees” Each element has a parent pointer ⇡ To compute the union ofExample set A with B, simply make B’s root the parent of A’s root. Figure 5.5 A directed-tree representation of two sets B,E and A, C, D, F, G, H . { } { } E H B C D F G A 18 / 33 Disjoint Set Operations Makeset(x), a procedure to form the set {x} Find(x), a procedure to find the representative element of the set containing x. -
Measure Theory
Measure Theory 1 Classes of Subsets 1.1 Topology Definition A Topological space (X; T ) consists of a non-empty set X to- gether with a collection T of subsets of X such that (T1) X; Á 2 T , T (T2) If A ; :::; A 2 T then n A 2 T , 1 n i=1 i S (T3) If Ai 2 T ; i 2 I for some index set I then i2I Ai 2 T . The elements of T are called the open sets. Note Every set has at least two topologies on it, namely the trivial ones T = fX; Ág and T = P(X), i.e. the collection of all subsets of X. T Lemma 1.1 If Tj; j 2 J are topologies on X then j2J Tj is a topology on X. Proof T (T1) X; Á 2 T for all j so X; Á 2 T . j j2J j T (T2) Take any finite collection of sets A ; :::; A 2 T : Then A ; :::; A T 1 n Tj2J j T 1 n 2 T for each j and so n A 2 T for each j and so n A 2 T . j i=1 i j T i=1 i j2J j (T3) Take any collection of sets A 2 T ; i 2 I. Then A 2 T ; for S i j2J j iS j all i 2 I and all j 2 J and so A 2 T for each j and thus A 2 T i2I i j i2I i j2J Tj. ¥ Corollary 1.2 There exists a topology U on R containing all intervals (a; b) and such that if T0 is any other topology containing all such intervals then U ⊆ T0.