8 Equivalence Relations

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8 Equivalence Relations 8.1 Relations 1. for sets A and B we define a relation from A to B to be a subset of A × B 2. for example the graph of a function is a relation from the domain (A) to its range (B). Note that every function a relation because it relates an element of A to an element of B. However not every relation a function because a relation may relate an element of A to more than just one element of B (or also because the empty relation is not a function). 3. a relation on a set A is a relation from A to A (i.e. subset of A × A, where A could be finite or infinite) 4. for a finite set A (with |A| = n), there are 2|A×A| = 2n2 possible relations on A, namely the elements of the powerset of A × A (note that the empty set is a relation on A as well) 8.2 Properties of Relations 1. a relation R defined on A is reflexive if ∀a ∈ A, then (a, a) ∈ R. 2. a relation R defined on A is symmetric if for elements a, b ∈ A we have that if (a, b) ∈ R then (b, a) ∈ R 3. a relation is transitive R defined on A if for elements a, b, c ∈ A we have that if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R 8.3 Equivalence Relations 1. a relation is an equivalence relation if it is reflexive, symmetric and transitive 2. if a relation R on a set A is an equivalence relation, then A can be partitioned into equivalence classes (note that the set A is partitioned, and the relation R helps find the partition, as the partition is unique for each relation) 3. if a relation R on a set A is an equivalence relation and a ∈ A, then the equivalence class [a] is given by [a] = {x ∈ A : xRa}, i.e. the equivalence class [a] contains a and all the elements related to a. 1 8.4 Properties of Equivalence Classes 1. if the class [a] has the same elements as the class [b], then the two classes are the same, i.e. [a] = [b]. It is always the case that two classes are either the same or they have no common elements (because the set of classes form a partition of A). This is Theorem 8.2. 2. since the equivalence classes produce a partition of A (see Theorem 8.3), we have the following: (1) no element of A belongs to two different equivalence classes, (2) every element of A belongs to an equivalence class, and (3) no equivalence class is empty. 3. also, the converse of the above item holds true: given a partition of a set A, there is always an equivalence relation R whose equivalence classes will be the elements of the partition. 8.5 Congruence Modulo n 1. recall divisibility definition: a|b if b = ak, ∃k ∈ Z 2. recall congruence modulo n definition: a ≡ b (mod n) if n|(a − b) 3. Theorem: Congruence modulo n is an equivalence relation on Z, and it has exactly n equivalence classes: [0], [1],..., [n − 1] (note that [0] = [n] = [2n] = ..., also [1] = [n + 1] = ...) 4. the classes above are given by the remainder of the given number when it’s divided by n 8.6 Integers Modulo n Skip this section. I will mention it briefly. 2.

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$Math Preliminaries$

Math Preliminaries

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$Math 296. Homework 3 (Due Jan 28) 1$

Math 296. Homework 3 (Due Jan 28) 1

Math 296. Homework 3 (due Jan 28) 1. Equivalence Classes. Let R be an equivalence relation on a set X. For each x ∈ X, consider the subset xR ⊂ X consisting of all the elements y in X such that xRy. A set of the form xR is called an equivalence class. (1) Show that xR = yR (as subsets of X) if and only if xRy. (2) Show that xR ∩ yR = ∅ or xR = yR. (3) Show that there is a subset Y (called equivalence classes representatives) of X such that X is the disjoint union of subsets of the form yR for y ∈ Y . Is the set Y uniquely determined? (4) For each of the equivalence relations from Problem Set 2, Exercise 5, Parts 3, 5, 6, 7, 8: describe the equivalence classes, find a way to enumerate them by picking a nice representative for each, and find the cardinality of the set of equivalence classes. [I will ask Ruthi to discuss this a bit in the discussion session.] 2. Pliability of Smooth Functions. This problem undertakes a very fundamental construction: to prove that ∞ −1/x2 C -functions are very soft and pliable. Let F : R → R be defined by F (x) = e for x 6= 0 and F (0) = 0. (1) Verify that F is infinitely differentiable at every point (don’t forget that you computed on a 295 problem set that the k-th derivative exists and is zero, for all k ≥ 1). −1/x2 ∞ (2) Let ϕ : R → R be defined by ϕ(x) = 0 for x ≤ 0 and ϕ(x) = e for x > 0.
Sets, Functions

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The Structure of the 3-Separations of 3-Connected Matroids

THE STRUCTURE OF THE 3{SEPARATIONS OF 3{CONNECTED MATROIDS JAMES OXLEY, CHARLES SEMPLE, AND GEOFF WHITTLE Abstract. Tutte deﬁned a k{separation of a matroid M to be a partition (A; B) of the ground set of M such that |A|; |B|≥k and r(A)+r(B) − r(M) <k. If, for all m<n, the matroid M has no m{separations, then M is n{connected. Earlier, Whitney showed that (A; B) is a 1{separation of M if and only if A is a union of 2{connected components of M.WhenM is 2{connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2{separations. When M is 3{connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3{ separations of M. 1. Introduction One of Tutte’s many important contributions to matroid theory was the introduction of the general theory of separations and connectivity [10] de- ﬁned in the abstract. The structure of the 1–separations in a matroid is elementary. They induce a partition of the ground set which in turn induces a decomposition of the matroid into 2–connected components [11]. Cun- ningham and Edmonds [1] considered the structure of 2–separations in a matroid. They showed that a 2–connected matroid M can be decomposed into a set of 3–connected matroids with the property that M can be built from these 3–connected matroids via a canonical operation known as 2–sum. Moreover, there is a labelled tree that gives a precise description of the way that M is built from the 3–connected pieces.
Matroids with Nine Elements

Matroids with nine elements Dillon Mayhew School of Mathematics, Statistics & Computer Science Victoria University Wellington, New Zealand [email protected] Gordon F. Royle School of Computer Science & Software Engineering University of Western Australia Perth, Australia [email protected] February 2, 2008 Abstract We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation conﬁrms and extends the results obtained in the 1960s by Blackburn, Crapo & Higgs. The matroids and associated data are stored in an online database, arXiv:math/0702316v1 [math.CO] 12 Feb 2007 and we give three short examples of the use of this database. 1 Introduction In the late 1960s, Blackburn, Crapo & Higgs published a technical report describing the results of a computer search for all simple matroids on up to eight elements (although the resulting paper [2] did not appear until 1973). In both the report and the paper they said 1 “It is unlikely that a complete tabulation of 9-point geometries will be either feasible or desirable, as there will be many thousands of them. The recursion g(9) = g(8)3/2 predicts 29260.” Perhaps this comment dissuaded later researchers in matroid theory, because their cata- logue remained unextended for more than 30 years, which surely makes it one of the longest standing computational results in combinatorics. However, in this paper we demonstrate that they were in fact unduly pessimistic, and describe an orderly algorithm (see McKay [7] and Royle [11]) that conﬁrms their computations and extends them by determining the 383172 pairwise non-isomorphic matroids on nine elements (see Table 1).
Section 3: Equivalence Relations

10.3.1 Section 3: Equivalence Relations • Definition: Let R be a binary relation on A. R is an equivalence relation on A if R is reflexive, symmetric, and transitive. • From the last section, we demonstrated that Equality on the Real Numbers and Congruence Modulo p on the Integers were reflexive, symmetric, and transitive, so we can describe them as equivalence relations. 10.3.2 Examples • What is the “smallest” equivalence relation on a set A? R = {(a,a) | a Î A}, so that n(R) = n(A). • What is the “largest” equivalence relation on a set A? R = A ´ A, so that n(R) = [n(A)]2. Equivalence Classes 10.3.3 • Definition: If R is an equivalence relation on a set A, and a Î A, then the equivalence class of a is defined to be: [a] = {b Î A | (a,b) Î R}. • In other words, [a] is the set of all elements which relate to a by R. • For example: If R is congruence mod 5, then [3] = {..., -12, -7, -2, 3, 8, 13, 18, ...}. • Another example: If R is equality on Q, then [2/3] = {2/3, 4/6, 6/9, 8/12, 10/15, ...}. • Observation: If b Î [a], then [b] = [a]. 10.3.4 A String Example • Let S = {0,1} and denote L(s) = length of s, for any string s Î S*. Consider the relation: R = {(s,t) | s,t Î S* and L(s) = L(t)} • R is an equivalence relation. Why? • REF: For all s Î S*, L(s) = L(s); SYM: If L(s) = L(t), then L(t) = L(s); TRAN: If L(s) = L(t) and L(t) = L(u), L(s) = L(u).
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Parity Systems and the Delta-Matroid Intersection Problem

Parity Systems and the Delta-Matroid Intersection Problem AndréBouchet ∗ and Bill Jackson † Submitted: February 16, 1998; Accepted: September 3, 1999. Abstract We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lovász to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two delta- matroids, or two orthogonal 2-polymatroids, on the same ground-sets. We show that many of the results of Lovászconcerning ‘double flowers’ and ‘projections’ carry over to parity systems. 1 Introduction: the delta-matroid intersec- tion problem A delta-matroid is a pair (V, ) with a finite set V and a nonempty collection of subsets of V , called theBfeasible sets or bases, satisfying the following axiom:B ∗Département d’informatique, Universitédu Maine, 72017 Le Mans Cedex, France. [email protected] †Department of Mathematical and Computing Sciences, Goldsmiths’ College, London SE14 6NW, England. [email protected] 1 the electronic journal of combinatorics 7 (2000), #R14 2 1.1 For B1 and B2 in and v1 in B1∆B2, there is v2 in B1∆B2 such that B B1∆ v1, v2 belongs to . { } B Here P ∆Q = (P Q) (Q P ) is the symmetric difference of two subsets P and Q of V . If X\ is a∪ subset\ of V and if we set ∆X = B∆X : B , then we note that (V, ∆X) is a new delta-matroid.B The{ transformation∈ B} (V, ) (V, ∆X) is calledB a twisting.
6. Localization

52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a ﬁeld, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can deﬁne addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
Computing Partitions Within SQL Queries: a Dead End? Frédéric Dumonceaux, Guillaume Raschia, Marc Gelgon

Computing Partitions within SQL Queries: A Dead End? Frédéric Dumonceaux, Guillaume Raschia, Marc Gelgon To cite this version: Frédéric Dumonceaux, Guillaume Raschia, Marc Gelgon. Computing Partitions within SQL Queries: A Dead End?. 2013. hal-00768156 HAL Id: hal-00768156 https://hal.archives-ouvertes.fr/hal-00768156 Submitted on 20 Dec 2012 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. Computing Partitions within SQL Queries: A Dead End? Frédéric Dumonceaux – Guillaume Raschia – Marc Gelgon December 20, 2012 Abstract The primary goal of relational databases is to provide efficient query processing on sets of tuples and thereafter, query evaluation and optimization strategies are a key issue in database implementation. Producing universally fast execu- tion plans remains a challenging task since the underlying relational model has a significant impact on algebraic definition of the operators, thereby on their implementation in terms of space and time complexity. At least, it should pre- vent a quadratic behavior in order to consider scaling-up towards the processing of large datasets. The main purpose of this paper is to show that there is no trivial relational modeling for managing collections of partitions (i.e.
$Introduction to Abstract Algebra (Math 113)$

Introduction to Abstract Algebra (Math 113)

Introduction to Abstract Algebra (Math 113) Alexander Paulin, with edits by David Corwin FOR FALL 2019 MATH 113 002 ONLY Contents 1 Introduction 4 1.1 What is Algebra? . 4 1.2 Sets . 6 1.3 Functions . 9 1.4 Equivalence Relations . 12 2 The Structure of + and × on Z 15 2.1 Basic Observations . 15 2.2 Factorization and the Fundamental Theorem of Arithmetic . 17 2.3 Congruences . 20 3 Groups 23 1 3.1 Basic Definitions . 23 3.1.1 Cayley Tables for Binary Operations and Groups . 28 3.2 Subgroups, Cosets and Lagrange's Theorem . 30 3.3 Generating Sets for Groups . 35 3.4 Permutation Groups and Finite Symmetric Groups . 40 3.4.1 Active vs. Passive Notation for Permutations . 40 3.4.2 The Symmetric Group Sym3 . 43 3.4.3 Symmetric Groups in General . 44 3.5 Group Actions . 52 3.5.1 The Orbit-Stabiliser Theorem . 55 3.5.2 Centralizers and Conjugacy Classes . 59 3.5.3 Sylow's Theorem . 66 3.6 Symmetry of Sets with Extra Structure . 68 3.7 Normal Subgroups and Isomorphism Theorems . 73 3.8 Direct Products and Direct Sums . 83 3.9 Finitely Generated Abelian Groups . 85 3.10 Finite Abelian Groups . 90 3.11 The Classification of Finite Groups (Proofs Omitted) . 95 4 Rings, Ideals, and Homomorphisms 100 2 4.1 Basic Definitions . 100 4.2 Ideals, Quotient Rings and the First Isomorphism Theorem for Rings . 105 4.3 Properties of Elements of Rings . 109 4.4 Polynomial Rings . 112 4.5 Ring Extensions . 115 4.6 Field of Fractions .
Partial Ordering Relations • a Relation Is Said to Be a Partial Ordering Relation If It Is Reflexive, Anti -Symmetric, and Transitive

Relations Chang-Gun Lee ([email protected]) Assistant Professor The School of Computer Science and Engineering Seoul National University Relations • The concept of relations is also commonly used in computer science – two of the programs are related if they share some common data and are not related otherwise. – two wireless nodes are related if they interfere each other and are not related otherwise – In a database, two objects are related if their secondary key values are the same • What is the mathematical definition of a relation? • Definition 13.1 (Relation): A relation is a set of ordered pairs – The set of ordered pairs is a complete listing of all pairs of objects that “satisfy” the relation •Examples: – Greater Than Re lat ion = {(2 {(21)(31)(32),1), (3,1), (3,2), ...... } – R = {(1,2), (1,3), (3,0)} (1,2)∈ R, 1 R 2 :"x is related by the relation R to y" Relations • Definition 13.2 (Relation on, between sets) Let R be a relation and let A and B be sets . –We say R is a relation on A provided R ⊆ A× A –We say R is a relation from A to B provided R ⊆ A× B Example Relations •Let A={1,2,3,4} and B={4,5,6,7}. Let – R={(11)(22)(33)(44)}{(1,1),(2,2),(3,3),(4,4)} – S={(1,2),(3,2)} – T={(1,4),(1,5),(4,7)} – U={(4,4),(5,2),(6,2),(7,3)}, and – V={(1,7),(7,1)} • All of these are relations – R is a relation on A.