AN AXIOMATIC APPROACH to DISTANCES BETWEEN CERTAIN DISCRETE STRUCTURES T. Margush a Dissertation Submitted to the Graduate Colle
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Relations-Handoutnonotes.Pdf
Introduction Relations Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Definition A binary relation from a set A to a set B is a subset R ⊆ A × B = {(a, b) | a ∈ A, b ∈ B} Spring 2006 Note the difference between a relation and a function: in a relation, each a ∈ A can map to multiple elements in B. Thus, Computer Science & Engineering 235 relations are generalizations of functions. Introduction to Discrete Mathematics Sections 7.1, 7.3–7.5 of Rosen If an ordered pair (a, b) ∈ R then we say that a is related to b. We [email protected] may also use the notation aRb and aRb6 . Relations Relations Graphical View To represent a relation, you can enumerate every element in R. A B Example a1 b1 a Let A = {a1, a2, a3, a4, a5} and B = {b1, b2, b3} let R be a 2 relation from A to B as follows: a3 b2 R = {(a1, b1), (a1, b2), (a1, b3), (a2, b1), a4 (a3, b1), (a3, b2), (a3, b3), (a5, b1)} b3 a5 You can also represent this relation graphically. Figure: Graphical Representation of a Relation Relations Reflexivity On a Set Definition Definition A relation on the set A is a relation from A to A. I.e. a subset of A × A. There are several properties of relations that we will look at. If the ordered pairs (a, a) appear in a relation on a set A for every a ∈ A Example then it is called reflexive. -
On the Satisfiability Problem for Fragments of the Two-Variable Logic
ON THE SATISFIABILITY PROBLEM FOR FRAGMENTS OF TWO-VARIABLE LOGIC WITH ONE TRANSITIVE RELATION WIESLAW SZWAST∗ AND LIDIA TENDERA Abstract. We study the satisfiability problem for two-variable first- order logic over structures with one transitive relation. We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model property nor the tree model property, to show decidability we introduce a novel model con- struction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decid- ability for the full two-variable logic with one transitive relation, hence contrary to our previous claim, [FO2 with one transitive relation is de- cidable, STACS 2013: 317-328], the status of the latter problem remains open. 1. Introduction The two-variable fragment of first-order logic, FO2, is the restriction of classical first-order logic over relational signatures to formulas with at most two variables. It is well-known that FO2 enjoys the finite model prop- erty [23], and its satisfiability (hence also finite satisfiability) problem is NExpTime-complete [7]. One drawback of FO2 is that it can neither express transitivity of a bi- nary relation nor say that a binary relation is a partial (or linear) order, or an equivalence relation. These natural properties are important for prac- arXiv:1804.09447v3 [cs.LO] 9 Apr 2019 tical applications, thus attempts have been made to investigate FO2 over restricted classes of structures in which some distinguished binary symbols are required to be interpreted as transitive relations, orders, equivalences, etc. -
Factorisations of Some Partially Ordered Sets and Small Categories
Factorisations of some partially ordered sets and small categories Tobias Schlemmer Received: date / Accepted: date Abstract Orbits of automorphism groups of partially ordered sets are not necessarily con- gruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that preserves the antisymmetry of the order Relation. Finally some suggestions are given, how the orbit categories can be represented by simple directed and annotated graphs and annotated binary relations. These relations are reflexive, and, in many cases, they can be chosen to be antisymmetric. From these constructions arise different suggestions for fundamental systems of partially ordered sets and reconstruction data which are illustrated by examples from mathematical music theory. Keywords ordered set · factorisation · po-group · small category Mathematics Subject Classification (2010) 06F15 · 18B35 · 00A65 1 Introduction In general, the orbits of automorphism groups of partially ordered sets cannot be considered as equivalence classes of a convenient congruence relation of the corresponding partial or- ders. If the orbits are not convex with respect to the order relation, the factor relation of the partial order is not necessarily a partial order. However, when we consider a partial order as a directed graph, the direction of the arrows is preserved during the factorisation in many cases, while the factor graph of a simple graph is not necessarily simple. Even if the factor relation can be used to anchor unfolding information [1], this structure is usually not visible as a relation. According to the common mathematical usage a fundamental system is a structure that generates another (larger structure) according to some given rules. -
Math 475 Homework #3 March 1, 2010 Section 4.6
Student: Yu Cheng (Jade) Math 475 Homework #3 March 1, 2010 Section 4.6 Exercise 36-a Let ͒ be a set of ͢ elements. How many different relations on ͒ are there? Answer: On set ͒ with ͢ elements, we have the following facts. ) Number of two different element pairs ƳͦƷ Number of relations on two different elements ) ʚ͕, ͖ʛ ∈ ͌, ʚ͖, ͕ʛ ∈ ͌ 2 Ɛ ƳͦƷ Number of relations including the reflexive ones ) ʚ͕, ͕ʛ ∈ ͌ 2 Ɛ ƳͦƷ ƍ ͢ ġ Number of ways to select these relations to form a relation on ͒ 2ͦƐƳvƷͮ) ͦƐ)! ġ ͮ) ʚ ʛ v 2ͦƐƳvƷͮ) Ɣ 2ʚ)ͯͦʛ!Ɛͦ Ɣ 2) )ͯͥ ͮ) Ɣ 2) . b. How many of these relations are reflexive? Answer: We still have ) number of relations on element pairs to choose from, but we have to 2 Ɛ ƳͦƷ ƍ ͢ include the reflexive one, ʚ͕, ͕ʛ ∈ ͌. There are ͢ relations of this kind. Therefore there are ͦƐ)! ġ ġ ʚ ʛ 2ƳͦƐƳvƷͮ)Ʒͯ) Ɣ 2ͦƐƳvƷ Ɣ 2ʚ)ͯͦʛ!Ɛͦ Ɣ 2) )ͯͥ . c. How many of these relations are symmetric? Answer: To select only the symmetric relations on set ͒ with ͢ elements, we have the following facts. ) Number of symmetric relation pairs between two elements ƳͦƷ Number of relations including the reflexive ones ) ʚ͕, ͕ʛ ∈ ͌ ƳͦƷ ƍ ͢ ġ Number of ways to select these relations to form a relation on ͒ 2ƳvƷͮ) )! )ʚ)ͯͥʛ )ʚ)ͮͥʛ ġ ͮ) ͮ) 2ƳvƷͮ) Ɣ 2ʚ)ͯͦʛ!Ɛͦ Ɣ 2 ͦ Ɣ 2 ͦ . d. -
PROBLEM SET THREE: RELATIONS and FUNCTIONS Problem 1
PROBLEM SET THREE: RELATIONS AND FUNCTIONS Problem 1 a. Prove that the composition of two bijections is a bijection. b. Prove that the inverse of a bijection is a bijection. c. Let U be a set and R the binary relation on ℘(U) such that, for any two subsets of U, A and B, ARB iff there is a bijection from A to B. Prove that R is an equivalence relation. Problem 2 Let A be a fixed set. In this question “relation” means “binary relation on A.” Prove that: a. The intersection of two transitive relations is a transitive relation. b. The intersection of two symmetric relations is a symmetric relation, c. The intersection of two reflexive relations is a reflexive relation. d. The intersection of two equivalence relations is an equivalence relation. Problem 3 Background. For any binary relation R on a set A, the symmetric interior of R, written Sym(R), is defined to be the relation R ∩ R−1. For example, if R is the relation that holds between a pair of people when the first respects the other, then Sym(R) is the relation of mutual respect. Another example: if R is the entailment relation on propositions (the meanings expressed by utterances of declarative sentences), then the symmetric interior is truth- conditional equivalence. Prove that the symmetric interior of a preorder is an equivalence relation. Problem 4 Background. If v is a preorder, then Sym(v) is called the equivalence rela- tion induced by v and written ≡v, or just ≡ if it’s clear from the context which preorder is under discussion. -
Section 4.1 Relations
Binary Relations (Donny, Mary) (cousins, brother and sister, or whatever) Section 4.1 Relations - to distinguish certain ordered pairs of objects from other ordered pairs because the components of the distinguished pairs satisfy some relationship that the components of the other pairs do not. 1 2 The Cartesian product of a set S with itself, S x S or S2, is the set e.g. Let S = {1, 2, 4}. of all ordered pairs of elements of S. On the set S x S = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), Let S = {1, 2, 3}; then (2, 4), (4, 1), (4, 2), (4, 4)} S x S = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2) , (3, 3)} For relationship of equality, then (1, 1), (2, 2), (3, 3) would be the A binary relation can be defined by: distinguished elements of S x S, that is, the only ordered pairs whose components are equal. x y x = y 1. Describing the relation x y if and only if x = y/2 x y x < y/2 For relationship of one number being less than another, we Thus (1, 2) and (2, 4) satisfy . would choose (1, 2), (1, 3), and (2, 3) as the distinguished ordered pairs of S x S. x y x < y 2. Specifying a subset of S x S {(1, 2), (2, 4)} is the set of ordered pairs satisfying The notation x y indicates that the ordered pair (x, y) satisfies a relation . -
Relations April 4
math 55 - Relations April 4 Relations Let A and B be two sets. A (binary) relation on A and B is a subset R ⊂ A × B. We often write aRb to mean (a; b) 2 R. The following are properties of relations on a set S (where above A = S and B = S are taken to be the same set S): 1. Reflexive: (a; a) 2 R for all a 2 S. 2. Symmetric: (a; b) 2 R () (b; a) 2 R for all a; b 2 S. 3. Antisymmetric: (a; b) 2 R and (b; a) 2 R =) a = b. 4. Transitive: (a; b) 2 R and (b; c) 2 R =) (a; c) 2 R for all a; b; c 2 S. Exercises For each of the following relations, which of the above properties do they have? 1. Let R be the relation on Z+ defined by R = f(a; b) j a divides bg. Reflexive, antisymmetric, and transitive 2. Let R be the relation on Z defined by R = f(a; b) j a ≡ b (mod 33)g. Reflexive, symmetric, and transitive 3. Let R be the relation on R defined by R = f(a; b) j a < bg. Symmetric and transitive 4. Let S be the set of all convergent sequences of rational numbers. Let R be the relation on S defined by f(a; b) j lim a = lim bg. Reflexive, symmetric, transitive 5. Let P be the set of all propositional statements. Let R be the relation on P defined by R = f(a; b) j a ! b is trueg. -
Relations II
CS 441 Discrete Mathematics for CS Lecture 22 Relations II Milos Hauskrecht [email protected] 5329 Sennott Square CS 441 Discrete mathematics for CS M. Hauskrecht Cartesian product (review) •Let A={a1, a2, ..ak} and B={b1,b2,..bm}. • The Cartesian product A x B is defined by a set of pairs {(a1 b1), (a1, b2), … (a1, bm), …, (ak,bm)}. Example: Let A={a,b,c} and B={1 2 3}. What is AxB? AxB = {(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)} CS 441 Discrete mathematics for CS M. Hauskrecht 1 Binary relation Definition: Let A and B be sets. A binary relation from A to B is a subset of a Cartesian product A x B. Example: Let A={a,b,c} and B={1,2,3}. • R={(a,1),(b,2),(c,2)} is an example of a relation from A to B. CS 441 Discrete mathematics for CS M. Hauskrecht Representing binary relations • We can graphically represent a binary relation R as follows: •if a R b then draw an arrow from a to b. a b Example: • Let A = {0, 1, 2}, B = {u,v} and R = { (0,u), (0,v), (1,v), (2,u) } •Note: R A x B. • Graph: 2 0 u v 1 CS 441 Discrete mathematics for CS M. Hauskrecht 2 Representing binary relations • We can represent a binary relation R by a table showing (marking) the ordered pairs of R. Example: • Let A = {0, 1, 2}, B = {u,v} and R = { (0,u), (0,v), (1,v), (2,u) } • Table: R | u v or R | u v 0 | x x 0 | 1 1 1 | x 1 | 0 1 2 | x 2 | 1 0 CS 441 Discrete mathematics for CS M. -
Chарtеr 2 RELATIONS
Ch=FtAr 2 RELATIONS 2.0 INTRODUCTION Every day we deal with relationships such as those between a business and its telephone number, an employee and his or her work, a person and other person, and so on. Relationships such as that between a program and a variable it uses and that between a computer language and a valid statement in this language often arise in computer science. The relationship between the elements of the sets is represented by a structure, called relation, which is just a subset of the Cartesian product of the sets. Relations are used to solve many problems such as determining which pairs of cities are linked by airline flights in a network or producing a useful way to store information in computer databases. In this chapter, we will study equivalence relation, equivalence class, composition of relations, matrix of relations, and closure of relations. 2.1 RELATION A relation is a set of ordered pairs. Let A and B be two sets. Then a relation from A to B is a subset of A × B. Symbolically, R is a relation from A to B iff R Í A × B. If (x, y) Î R, then we can express it by writing xRy and say that x is related to y with relation R. Thus, (x, y) Î R Û xRy 2.2 RELATION ON A SET A relation R on a set A is the subset of A × A, i.e., R Í A × A. Here both the sets A and B are same. -xample Let A = {2, 3, 4, 5} and B = {2, 4, 6, 10, 12}. -
Relations CIS002-2 Computational Alegrba and Number Theory
Relations CIS002-2 Computational Alegrba and Number Theory David Goodwin [email protected] 11:00, Tuesday 29th November 2011 bg=whiteRelations Equivalence Relations Class Exercises Outline 1 Relations Inverse Relation Composition Reflexive relation 2 Equivalence Symmetric relation Relations Antisymmetric relation Equivalence relation Transitive relation Equivalence classes Partial order 3 Class Exercises bg=whiteRelations Equivalence Relations Class Exercises Outline 1 Relations Inverse Relation Composition Reflexive relation 2 Equivalence Symmetric relation Relations Antisymmetric relation Equivalence relation Transitive relation Equivalence classes Partial order 3 Class Exercises bg=whiteRelations Equivalence Relations Class Exercises Relations A (binary) relation R from a set X to a set Y is a subset of the Cartesian product X × Y . If (x; y 2 R, we write xRy and say that x is related to y. If X = Y , we call R a (binary) relation on X . A function is a special type of relation. A function f from X to Y is a relation from X to Y having the properties: • The domain of f is equal to X . • For each x 2 X , there is exactly one y 2 Y such that (x; y) 2 f bg=whiteRelations Equivalence Relations Class Exercises Relations - example Let X = f2; 3; 4g and Y = f3; 4; 5; 6; 7g If we define a relation R from X to Y by (x; y) 2 R if x j y we obtain R = f(2; 4); (2; 6); (3; 3); (3; 6); (4; 4)g bg=whiteRelations Equivalence Relations Class Exercises Relations - reflexive A relation R on a set X is reflexive if (x; x) 2 X , if (x; y) 2 R for all x 2 X . -
CDM Relations
CDM 1 Relations Relations Operations and Properties Orders Klaus Sutner Carnegie Mellon University Equivalence Relations 30-relations 2017/12/15 23:22 Closures Relations 3 Binary Relations 4 We have seen how to express general concepts (or properties) as sets: we form the set of all objects that “fall under” the concept (in Frege’s Let’s focus on the binary case where two objects are associated, though terminology). Thus we can form the set of natural numbers, of primes, of not necessarily from the same domain. reals, of continuous functions, of stacks, of syntactically correct The unifying characteristic is that we have some property (attribute, C-programs and so on. quality) that can hold or fail to hold of any two objects from the Another fundamental idea is to consider relationships between two or appropriate domains. more objects. Here are some typical examples: Standard notation: for a relation P and suitable objects a and b write P (a, b) for the assertion that P holds on a and b. divisibility relation on natural numbers, Thus we can associate a truth value with P (a, b): the assertion holds if a less-than relation on integers, and b are indeed related by P , and is false otherwise. greater-than relation on rational numbers, For example, if P denotes the divisibility relation on the integers then the “attends course” relation for students and courses, P (3, 9) holds whereas P (3, 10) is false. the “is prerequisite” relation for courses, Later we will be mostly interested in relations where we can effectively the “is a parent of” relation for humans, determine whether P (a, b) holds, but for the time being we will consider the “terminates on input and produces output” relation for programs the general case. -
Residuated Relational Systems We Begin by Introducing the Central Notion That Will Be Used Through- out the Paper
RESIDUATED RELATIONAL SYSTEMS S. BONZIO AND I. CHAJDA Abstract. The aim of the present paper is to generalize the con- cept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjoint- ness in such a way that the ordering relation can be harmlessly replaced by a binary relation. By enriching such binary relation with additional properties we get interesting properties of residu- ated relational systems which are analogical to those of residuated posets and lattices. 1. Introduction The study of binary relations traces back to the work of J. Riguet [14], while a first attempt to provide an algebraic theory of relational systems is due to Mal’cev [12]. Relational systems of different kinds have been investigated by different authors for a long time, see for example [4], [3], [8], [9], [10]. Binary relational systems are very im- portant for the whole of mathematics, as relations, and thus relational systems, represent a very general framework appropriate for the de- scription of several problems, which can turn out to be useful both in mathematics and in its applications. For these reasons, it is fundamen- tal to study relational systems from a structural point of view. In order to get deeper results meeting possible applications, we claim that the usual domain of binary relations shall be expanded. More specifically, arXiv:1810.09335v1 [math.LO] 22 Oct 2018 our aim is to study general binary relations on an underlying algebra whose operations interact with them.