Mathematical Appendix
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Lecture 1 Mathematical Preliminaries
Lecture 1 Mathematical Preliminaries A. Banerji July 26, 2016 (Delhi School of Economics) Introductory Math Econ July 26, 2016 1 / 27 Outline 1 Preliminaries Sets Logic Sets and Functions Linear Spaces (Delhi School of Economics) Introductory Math Econ July 26, 2016 2 / 27 Preliminaries Sets Basic Concepts Take as understood the notion of a set. Usually use upper case letters for sets and lower case ones for their elements. Notation. a 2 A, b 2= A. A ⊆ B if every element of A also belongs to B. A = B if A ⊆ B and B ⊆ A are both true. A [ B = fxjx 2 A or x 2 Bg. The ‘or’ is inclusive of ‘both’. A \ B = fxjx 2 A and x 2 Bg. What if A and B have no common elements? To retain meaning, we invent the concept of an empty set, ;. A [; = A, A \; = ;. For the latter, note that there’s no element in common between the sets A and ; because the latter does not have any elements. ; ⊆ A, for all A. Every element of ; belongs to A is vacuously true since ; has no elements. This brings us to some logic. (Delhi School of Economics) Introductory Math Econ July 26, 2016 3 / 27 Preliminaries Logic Logic Statements or propositions must be either true or false. P ) Q. If the statement P is true, then the statement Q is true. But if P is false, Q may be true or false. For example, on real numbers, let P = x > 0 and Q = x2 > 0. If P is true, so is Q, but Q may be true even if P is not, e.g. -
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. -
Political Game Theory Nolan Mccarty Adam Meirowitz
Political Game Theory Nolan McCarty Adam Meirowitz To Liz, Janis, Lachlan, and Delaney. Contents Acknowledgements vii Chapter 1. Introduction 1 1. Organization of the Book 2 Chapter 2. The Theory of Choice 5 1. Finite Sets of Actions and Outcomes 6 2. Continuous Outcome Spaces* 10 3. Utility Theory 17 4. Utility representations on Continuous Outcome Spaces* 18 5. Spatial Preferences 19 6. Exercises 21 Chapter 3. Choice Under Uncertainty 23 1. TheFiniteCase 23 2. Risk Preferences 32 3. Learning 37 4. Critiques of Expected Utility Theory 41 5. Time Preferences 46 6. Exercises 50 Chapter 4. Social Choice Theory 53 1. The Open Search 53 2. Preference Aggregation Rules 55 3. Collective Choice 61 4. Manipulation of Choice Functions 66 5. Exercises 69 Chapter 5. Games in the Normal Form 71 1. The Normal Form 73 2. Solutions to Normal Form Games 76 3. Application: The Hotelling Model of Political Competition 83 4. Existence of Nash Equilibria 86 5. Pure Strategy Nash Equilibria in Non-Finite Games* 93 6. Application: Interest Group Contributions 95 7. Application: International Externalities 96 iii iv CONTENTS 8. Computing Equilibria with Constrained Optimization* 97 9. Proving the Existence of Nash Equilibria** 98 10. Strategic Complementarity 102 11. Supermodularity and Monotone Comparative Statics* 103 12. Refining Nash Equilibria 108 13. Application: Private Provision of Public Goods 109 14. Exercises 113 Chapter 6. Bayesian Games in the Normal Form 115 1. Formal Definitions 117 2. Application: Trade restrictions 119 3. Application: Jury Voting 121 4. Application: Jury Voting with a Continuum of Signals* 123 5. Application: Public Goods and Incomplete Information 126 6. -
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 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 . -
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. -
Functions, Relations, Partial Order Relations Definition: the Cartesian Product of Two Sets a and B (Also Called the Product
Functions, Relations, Partial Order Relations Definition: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all pairs (a,b) where a ∈ A and b ∈ B . It is denoted A X B. Example : A ={1, 2), B= { a, b} A X B = { (1, a), (1, b), (2, a), (2, b)} Solve the following: X = {a, c} and Y = {a, b, e, f}. Write down the elements of: (a) X × Y (b) Y × X (c) X 2 (= X × X) (d) What could you say about two sets A and B if A × B = B × A? Definition: A function is a subset of the Cartesian product.A relation is any subset of a Cartesian product. For instance, a subset of , called a "binary relation from to ," is a collection of ordered pairs with first components from and second components from , and, in particular, a subset of is called a "relation on ." For a binary relation , one often writes to mean that is in . If (a, b) ∈ R × R , we write x R y and say that x is in relation with y. Exercise 2: A chess board’s 8 rows are labeled 1 to 8, and its 8 columns a to h. Each square of the board is described by the ordered pair (column letter, row number). (a) A knight is positioned at (d, 3). Write down its possible positions after a single move of the knight. Answer: (b) If R = {1, 2, ..., 8}, C = {a, b, ..., h}, and P = {coordinates of all squares on the chess board}, use set notation to express P in terms of R and C. -
Introduction to Relations Binary Relation
Introduction to Relations CSE 191, Class Note 09 Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Descrete Structures 1 / 57 Binary relation Definition: Let A and B be two sets. A binary relation from A to B is a subset of A B. × In other words, a binary relation from A to B is a set of pair (a, b) such that a A and b B. ∈ ∈ We often omit “binary” if there is no confusion. If R is a relation from A to B and (a, b) R, we say a is related to b by R. ∈ We write a R b. Example: Let A be the set of UB students, and B be the set of UB courses. Define relation R from A to B as follows: a R b if and only if student a takes course b. So for every student x in this classroom, we have that (x, CSE191) R, or ∈ equivalently, x R CSE191. For your TA Ding , we know that Ding is NOT related to CSE191 by R. c XinSo He (University(Ding, CSE at Buffalo)191) R. CSE 191 Descrete Structures 3 / 57 6∈ Relation on a set We are particularly interested in binary relations from a set to the same set. If R is a relation from A to A, then we say R is a relation on set A. Example: We can define a relation R on the set of positive integers such that a R b if and only if a b. | 3 R 6. -
Mereology and the Sciences, Berlin: Springer, 2014, Pp
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilPapers [In Claudio Calosi and Pierluigi Graziani (eds.), Mereology and the Sciences, Berlin: Springer, 2014, pp. 359–370] Appendix. Formal Theories of Parthood Achille C. Varzi This Appendix gives a brief overview of the main formal theories of parthood, or mereologies, to be found in the literature.1 The focus is on classical theories, so the survey is not meant to be exhaustive. Moreover, it does not cover the many philo- sophical issues relating to the endorsement of the theories themselves, concerning which the reader is referred to the Selected Bibliography at the end of the volume. In particular, we shall be working under the following simplifying assumptions:2 — Absoluteness: Parthood is a two-place relation; it does not hold relative to time, space, spacetime regions, sortals, worlds, or anything else.3 — Monism: There is a single relation of parthood that applies to every entity inde- pendently of its ontological category.4 — Precision: Parthood is not a source of vagueness: there is always a fact of the mat- ter as to whether the parthood relation obtains between any given pair of things.5 For definiteness, all theories will be formulated in a standard first-order language with identity, supplied with a distinguished binary predicate constant, ‘P’, to be in- terpreted as the parthood relation. The underlying logic will be the classical predi- cate calculus. 1 Core Principles As a minimal requirement on ‘P’, it is customary to assume that it stands for a partial order—a reflexive, transitive, and antisymmetric relation:6 (P.1) Pxx Reflexivity (P.2) (Pxy ∧ Pyz) → Pxz Transitivity (P.3) (Pxy ∧ Pyx) → x = y Antisymmetry Together, these three axioms are meant to fix the intended meaning of the parthood predicate. -
Almost Lattices
JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4866, ISSN (o) 2303-4947 www.imvibl.org /JOURNALS / JOURNAL Vol. 9(2019), 155-171 DOI: 10.7251/JIMVI1901155R Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) ALMOST LATTICES G. Nanaji Rao and Habtamu Tiruneh Alemu Abstract. The concept of an Almost Lattice (AL) is introduced and given certain examples of an AL which are not lattices. Also, some basic properties of an AL are proved and set of equivalent conditions are established for an AL to become a lattice. Further, the concept of AL with 0 is introduced and some basic properties of an AL with 0 are proved. 1. Introduction It was Garett Birkhoff's (1911 - 1996) work in the mid thirties that started the general development of the lattice theory. In a brilliant series of papers, he demonstrated the importance of the lattice theory and showed that it provides a unified frame work for unrelated developments in many mathematical disciplines. V. Glivenko, Karl Menger, John Van Neumann, Oystein Ore, George Gratzer, P. R. Halmos, E. T. Schmidt, G. Szasz, M. H. Stone , R. P. Dilworth and many others have developed enough of this field for making it attractive to the mathematicians and for its further progress. The traditional approach to lattice theory proceeds from partially ordered sets to general lattices, semimodular lattices, modular lat- tices and finally to distributive lattices. In this paper, we introduced the concept of an Almost lattice AL which is a generalization of a lattice and we gave certain examples of ALs which are not lattices.