DOCSLIB.ORG

Relations Department of Computer Sciences College of Engineering Florida Tech Fall 2011

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Relations Department of Computer Sciences College of Engineering Florida Tech Fall 2011 CSE 1400 Applied Discrete Mathematics Relations Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Relations and Their Graphs 1 A Relation’s Domain, Co-domain, and Range 2 A Sampling of Relations 3 Equality 3 Less than 3 Divides 3 Congruence Modulo n 4 Perpendicular on Lines 5 The Incestuous and Empty Relations 5 A Relation is a Set of Ordered Pairs 6 The Inverse Relation 6 Counting Relations 7 Relational Properties 7 Reflexive Property 7 Symmetric Property 8 Antisymmetric Property 10 Transitive Property 11 Orders and Equivalences 11 Orders 12 Well-Ordered Sets 13 Equivalences 13 Equivalence Relations Partition a Set 14 Stirling Numbers of the Second Kind 16 cse 1400 applied discrete mathematics relations 2 Problems on Relations 18 Abstract A relation ∼ describes how things are connected. That a thing a is related to a thing b can be represented by 1. An ordered pair (a, b). • 2. An directed edge a •b . 3. Or more commonly, simply using relational notation a ∼ b. A relation is 1. A set G of ordered pairs. 2. A directed graph G of nodes and edges. 3. A matrix of True and False values. Higher-dimensional relations among a, b, c or more parameters can be defined. Higher-dimensional relations occur as tables in relational databases and as data in multi-variable problems. Relations and Their Graphs A relation is a set of ordered pairs. A relation can be pictured of as a graph G. •g G = f(x, y) : x ∼ yg = f(x, y) : x is related to y.g h• There are many examples of relations. You are, no doubt, familiar d• c• with relations among people: Mother-Daughter, Father-Son, Parent- Child, Aunt-Nephew. Familial relations often become murky. We study relations that can be precisely defined. A few common rela- •e • f tions are equality, congruence mod n, less than, divides, subset, and •a perpendicular. •b This In this course, relationships will be between two things a and b. graph represents the relation Relationships among 3 or more things are common and useful, but G = f(a, b), (b, c), (c, d), (d, h), (h, f ), ( f , e), (e, a)g these ideas are not within the scope of this course. The course studies binary relations. A Relation’s Domain, Co-domain, and Range The things involved in a relation need names:Call them the things x and y. Write x ∼ y to express the phrase “x is related to y.” cse 1400 applied discrete mathematics relations 3 The value x belongs to a set X called the domain of ∼. The value y belongs to a set Y called the co-domain of ∼. The domain X is the set of elements that appear on the left-hand side of ∼. For this course, you can assume that every element in X appears on the left-hand side of ∼, that is, every relation we en- counter is total, defined on all members of X. A relation ∼ is said to be total when The co-domain Y is the set of elements that can appear on the every x 2 X occurs on on the left-hand side of ∼ for some y 2 Y. right-hand side of ∼. A potentially smaller set is the range of ∼. (8x 2 X)(9y 2 Y)(x ∼ y) The range is the set of elements in Y that actually do appear on the right-hand side of ∼. In general, not every element y in Y occurs If there is an x 2 X such that no y 2 Y is related to x, then ∼ is said to be on the right-hand side of ∼. A relation is into its co-domain and partial. onto its range. That is, the co-domain of a relation, is the set of (9x 2 X)(8y 2 Y)(x 6∼ y) values y that could be related to some x. When a relation is onto its An instance of a partial relation on the co-domain every element in Y is related to some element x in X, real numbers is the pairing (x, 1/x), written which is undefined (has a pole) at x = 0. (8y 2 Y)(9x 2 X)(x ∼ y) A relation maps an element x in the domain X onto zero or more elements y in the range A ⊆ Y. A relation maps its domain into its Consider the graph in figure 1 that depicts a relation from vertices co-domain and onto its range. x0, x1, x2, x3 in domain X to vertices y0, y1, y2, y3 in co-domain Y X Y Figure 1:Domain X = fx0, x1, x2, x3g, Co-Domain Y = fy , y , y , y g, x3 y3 0 1 2 3 Range A = fy0, y1, y3g, x2 y2 x1 y1 x0 y0 Here are some things to notice about figure 1. 1. The relation is total: there is at least one directed edge from each vertex in X. 2. The relation is not onto its co-domain. 3. The relation is non-deterministic: A given input x can be related to many outputs y. For instance, in figure 1 x0 is related to y0 and y1. A relation is partial when a given input x cannot be related to any output y. cse 1400 applied discrete mathematics relations 4 X Y Figure 2: A partial relation: The relation is not defined on x . x3 y3 1 x2 y2 x1 y1 x0 y0 A Sampling of Relations You are familiar with many mathematical relations: Equality, less than, multiple of, and so on. These relations are between two things : a and b, and are called binary relations. Multidimensional relations are common in practice. Equality Equality is the most basic relation. 25 −10 72 5 = = = 5 −2 72 5 equal :: Integer -> Integer -> Bool equal a b = (0 == b - a) The name (5, 25/5, . .) can change, the thing remains the same. Figure 3: Haskell for deciding equality on the integers. Less than lessthan :: Integer -> Integer -> Bool Less than establishes an order on the integers. lessthan a b = (0 < b - a) Figure 4: Haskell for deciding less than ··· < −3 < −2 < −1 < 0 < 1 < 2 < 3 ··· on the integers. Divides 7 divides 35. “b divides a” is written b j a A natural number b divides a natural number a when “b does not divide a” is written there is an quotient q 2 N such a = bq. Divides is a rela- tion on N × N. b - a For instance, 15 divides 60 because “a is a multiple of b” is written a ≡ 0 mod b 60 = 15 · 4 “a is not a multiple of b” is written Stated the other way around, 60 is a multiple of 15. On the other a 6≡ 0 mod b hand, 60 is not a multiple of 8; 8 does not divide 60. 60 divided by 8 leaves a remainder of 4. In general, let a and b be natural numbers. Then b divides a, if there is a natural number q such that a = bq. If b divides a, then a is a multiple of b. cse 1400 applied discrete mathematics relations 5 The natural numbers can be partially ordered by the divides rela- 7 tion. The Hesse graph in figure 5 illustrates this ordering for the first 3 6 9 few natural numbers. Given natural numbers a and b, it is straightforward to write code 1 2 4 8 0 to test if a divides b. 5 10 Congruence Modulo n 11 Figure 5: 1 is the smallest natural number with respect to divides; 1 divides every natural number. 0 is the Congruence mod n has applications in many areas, cryp- largest natural number with respect to tology is just one interesting application that can be divides; every natural number divides 0. named. When natural number n divides the difference a − b, the integers a and b are said to be congruent mod n. For instance, the following pairs are congruent mod 2. (15, 9), (26, 30), (−9, 21), (6, −28), (17, −17) divides :: Nat -> Nat -> Bool while the following pairs are congruent mod 3. divides a b = (a ’div’ b == 0) ( ) ( ) (− ) ( − ) ( − ) Figure 6: Haskell for deciding divides 15, 9 , 26, 29 , 8, 19 , 6, 27 , 17, 16 on the natural numbers. n is called the modulus. Congruence collects pairs of integers based on their remainder when divided by n. Mod 2 collects pairs (a, b) that are both even (2s, 2t) or both odd (2s + 1, 2t + 1). Congruence mod 2 partitions the integers into two If both a and b are even, then their equivalence classes: The even integers and the odd integers. difference is divisible by 2. If both a and b are odd, then their difference is Mod 3 collects pairs (a, b) based on remainders upon division by divisible by 2. 3. There are 3 cases. 1. Both a and b are multiples of three: (3s, 3t) 2. Both have a remainder of 1 when divided by three: (3s + 1, 3t + 1) 3. Both have a remainder of 2 when divided by three: (3s + 2, 3t + 2) Congruence mod 3 partitions the integers into three equivalence classes: Integers that are multiples of 3, integers that are multiples of 3 plus 1, and integers that are multiples of 3 plus 2. In general, two integers a and b are congruent mod n if a − b is a multiple of n. Congruence mod n partitions the integers into n equivalence classes. A convenient notation for these sets of integers is [0], [1], [2],..., [n − 1] where syntatic sugar can be added to denote [r] = fnk + r : k 2 Zg the modulus: [r]n.
Load more

Recommended publications
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • 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.
    [Show full text]
  • Symmetric Relations and Cardinality-Bounded Multisets in Database Systems
    Symmetric Relations and Cardinality-Bounded Multisets in Database Systems Kenneth A. Ross Julia Stoyanovich Columbia University¤ [email protected], [email protected] Abstract 1 Introduction A relation R is symmetric in its ¯rst two attributes if R(x ; x ; : : : ; x ) holds if and only if R(x ; x ; : : : ; x ) In a binary symmetric relationship, A is re- 1 2 n 2 1 n holds. We call R(x ; x ; : : : ; x ) the symmetric com- lated to B if and only if B is related to A. 2 1 n plement of R(x ; x ; : : : ; x ). Symmetric relations Symmetric relationships between k participat- 1 2 n come up naturally in several contexts when the real- ing entities can be represented as multisets world relationship being modeled is itself symmetric. of cardinality k. Cardinality-bounded mul- tisets are natural in several real-world appli- Example 1.1 In a law-enforcement database record- cations. Conventional representations in re- ing meetings between pairs of individuals under inves- lational databases su®er from several consis- tigation, the \meets" relationship is symmetric. 2 tency and performance problems. We argue that the database system itself should pro- Example 1.2 Consider a database of web pages. The vide native support for cardinality-bounded relationship \X is linked to Y " (by either a forward or multisets. We provide techniques to be im- backward link) between pairs of web pages is symmet- plemented by the database engine that avoid ric. This relationship is neither reflexive nor antire- the drawbacks, and allow a schema designer to flexive, i.e., \X is linked to X" is neither universally simply declare a table to be symmetric in cer- true nor universally false.
    [Show full text]
  • On Effective Representations of Well Quasi-Orderings Simon Halfon
    On Effective Representations of Well Quasi-Orderings Simon Halfon To cite this version: Simon Halfon. On Effective Representations of Well Quasi-Orderings. Other [cs.OH]. Université Paris-Saclay, 2018. English. NNT : 2018SACLN021. tel-01945232 HAL Id: tel-01945232 https://tel.archives-ouvertes.fr/tel-01945232 Submitted on 5 Dec 2018 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. On Eective Representations of Well asi-Orderings ese` de doctorat de l’Universite´ Paris-Saclay prepar´ ee´ a` l’Ecole´ Normale Superieure´ de Cachan au sein du Laboratoire Specication´ & Verication´ Present´ ee´ et soutenue a` Cachan, le 29 juin 2018, par Simon Halfon Composition du jury : Dietrich Kuske Rapporteur Professeur, Technische Universitat¨ Ilmenau Peter Habermehl Rapporteur Maˆıtre de Conferences,´ Universite´ Paris-Diderot Mirna Dzamonja Examinatrice Professeure, University of East Anglia Gilles Geeraerts Examinateur Associate Professor, Universite´ Libre de Bruxelles Sylvain Conchon President´ du Jury Professeur, Universite´ Paris-Sud Philippe Schnoebelen Directeur de these` Directeur de Recherche, CNRS Sylvain Schmitz Co-encadrant de these` Maˆıtre de Conferences,´ ENS Paris-Saclay ` ese de doctorat ED STIC, NNT 2018SACLN021 Acknowledgements I would like to thank the reviewers of this thesis for their careful proofreading and pre- cious comment.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • SNAC: an Unbiased Metric Evaluating Topology Recognize Ability of Network Alignment
    SNAC: An Unbiased Metric Evaluating Topology Recognize Ability of Network Alignment Hailong Li, Naiyue Chen* School of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, China [email protected], [email protected] ABSTRACT Network alignment is a problem of finding the node mapping between similar networks. It links the data from separate sources and is widely studied in bioinformation and social network fields. The critical difference between network alignment and exact graph matching is that the network alignment considers node mapping in non-isomorphic graphs with error tolerance. Researchers usually utilize AC (accuracy) to measure the performance of network alignments which comparing each output element with the benchmark directly. However, this metric neglects that some nodes are naturally indistinguishable even in single graphs (e.g., nodes have the same neighbors) and no need to distinguish across graphs. Such neglect leads to the underestimation of models. We propose an unbiased metric for network alignment that takes indistinguishable nodes into consideration to address this problem. Our detailed experiments with different scales on both synthetic and real-world datasets demonstrate that the proposed metric correctly reflects the deviation of result mapping from benchmark mapping as standard metric AC does. Comparing with the AC, the proposed metric effectively blocks the effect of indistinguishable nodes and retains stability under increasing indistinguishable nodes. Keywords: metric; network alignment; graph automorphism; symmetric nodes. * Corresponding author 1. INTRODUCTION Network, or graph1, can represent complex relationships of objects (e.g., article reference relationship, protein-protein interaction) because of its flexibility. However, flexibility sometimes exhibits an irregular side, leading to some problems related to graph challenges.
    [Show full text]
  • Mereology Then and Now
    Logic and Logical Philosophy Volume 24 (2015), 409–427 DOI: 10.12775/LLP.2015.024 Rafał Gruszczyński Achille C. Varzi MEREOLOGY THEN AND NOW Abstract. This paper offers a critical reconstruction of the motivations that led to the development of mereology as we know it today, along with a brief description of some questions that define current research in the field. Keywords: mereology; parthood; formal ontology; foundations of mathe- matics 1. Introduction Understood as a general theory of parts and wholes, mereology has a long history that can be traced back to the early days of philosophy. As a formal theory of the part-whole relation or rather, as a theory of the relations of part to whole and of part to part within a whole it is relatively recent and came to us mainly through the writings of Edmund Husserl and Stanisław Leśniewski. The former were part of a larger project aimed at the development of a general framework for formal ontology; the latter were inspired by a desire to provide a nominalistically acceptable alternative to set theory as a foundation for mathematics. (The name itself, ‘mereology’ after the Greek word ‘µρoς’, ‘part’ was coined by Leśniewski [31].) As it turns out, both sorts of motivation failed to quite live up to expectations. Yet mereology survived as a theory in its own right and continued to flourish, often in unexpected ways. Indeed, it is not an exaggeration to say that today mereology is a central and powerful area of research in philosophy and philosophical logic. It may be helpful, therefore, to take stock and reconsider its origins.
    [Show full text]
  • 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 .
    [Show full text]
  • 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.
    [Show full text]
  • Binary Relations and Preference Modeling
    Chapter 2 Binary Relations and Preference Modeling 2.1. Introduction This volume is dedicated to concepts, results, procedures and software aiming at helping people make a decision. It is then natural to investigate how the various courses of action that are involved in this decision compare in terms of preference. The aim of this chapter is to propose a brief survey of the main tools and results that can be useful to do so. The literature on preference modeling is vast. This can first be explained by the fact that the question of modeling preferences occurs in several disciplines, e.g. – in Economics, where one tries to model the preferences of a ‘rational consumer’ [e.g. DEB 59]; – in Psychology in which the study of preference judgments collected in experi- ments is quite common [KAH 79, KAH 81]; – in Political Sciences in which the question of defining a collective preference on the basis of the opinion of several voters is central [SEN 86]; – in Operational Research in which optimizing an objective function implies the definition of a direction of preference [ROY 85]; and – in Artificial Intelligence in which the creation of autonomous agents able to take decisions implies the modeling of their vision of what is desirable and what is less so [DOY 92]. Chapter written by Denis BOUYSSOU and Philippe VINCKE. 49 50 Decision Making Moreover, the question of preference modeling can be studied from a variety of perspectives [BEL 88], including: –a normative perspective, where one investigates preference models that are likely to lead to a ‘rational behavior’; –a descriptive perspective, in which adequate models to capture judgements ob- tained in experiments are sought; or –a prescriptive perspective, in which one tries to build a preference model that is able to lead to an adequate recommendation.
    [Show full text]
  • 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}.
    [Show full text]