DOCSLIB.ORG

Relations and Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

2 Relations and Functions 2.1 INTRODUCTION The word relation implies an association of two objects, two persons, etc., according to some property possessed by them, e.g. (i) A is the son of B. The relation R here is “is the son of ”. (ii) Delhi is the capital of INDIA. The relation R here is “is the capital of ”. In Mathematics we can study relationship between a real number and a number which is smaller than it, between a program and a variable it uses. Relationship between elements of sets is represented using a structure called relation. 2.2 RELATION OR BINARY RELATION Let A and B be two non-empty sets. A binary relation or simply a relation from A to B is a sub- set of A × B. Given x ∈A and y ∈ B, we write x R y if (x, y) ∈ R and x R y if (x, y) ∉ R. If R is relation from A to A, then R is said to be relation on A. A binary relation on a set A is a subset of A × A. This definition needs some explanation, since the connection between it and the informal idea of a relation might not be obvious. Note: The word ‘binary’ refers to the fact that the relation is between two elements of A. As this is the only kind of relation we will be studying, we will omit the word ‘binary’ in what follows. The easiest way to do this is by means of an example. Let A = {1, 3, 5} be a set. Then A × A = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}. Unit2.indd 29 9/15/2009 3:48:24 PM Relations and Functions • 31 3. If x > y. The relation x R3 y means the ordered pairs in which the first element is greater than the second. These are (3, 1), (5, 1), and (5, 3). ∴ R3 = {(3, 1), (5, 1), (5, 3)}, a subset of A × A and so is a relation of A. ABU 1 1 3 3 5 5 Hence, in all the above cases, we have R1 ⊆ A × A, R2 ⊆ A × A, and R3 ⊆ A × A. Thus a relation R on any set is any subset of the Cartesian product A × A. 2.3 DOMAIN AND RANGE OF RELATION For a relation R, domain is a set of elements in A which are related to some element in B. Similarly range of R is a set of elements in B which are related to some element in A. Example 1: If A = {a, b, c}, B = {x, y, z}; and R = {(a, x), (a, z), (b, y), (c, x)}. Find the domain and range of R. Solution Domain of R = the set of first components of the ordered pairs in R = d(R) = {a, b, c} Range of R = the set of second components of the ordered pairs in R = r(R) = {x, y, z}. Example 2: Find the number of relations from A = {a, b, c} to B = {x, y, z}. Solution There are 3 × 3 = 9 elements in A × B and hence there are m = 29 = 512 subsets of A × B. Thus there are m = 512 relations from A × B Unit2.indd 31 9/15/2009 3:48:25 PM 34 • A Textbook of Discrete Mathematics Let A = {1, 2, 3} and B = {5, 6} R = {(1, 5), (1, 6), (2, 6), (3, 6)} 56 1 11 2 01 3 01 2.7 COMPOSITION OF RELATIONS Let A, B, C be three sets and a relation R1 from the set A to B be a R1 b where a ∈ A and b ∈ B, and the relation R2 from B to C be b R2 c where b ∈ B and c ∈ C. Then the composition of the relation R1 and R2 denoted by R1 o R2 or R1R2 is a relation from A to C defined by R1 o R2 = {(a, c): b ∈ B is an element such that aRb and bRc}, where a ∈ A and c ∈ C. Computing the composite of two relations requires that we find elements that are the sec- ond element of ordered pairs in first relation and the first element of ordered pairs in the second relation. Example 5: Let A = {1, 3, 5}, Let R be the relation such that xRy, iff y = x + 2 and S be the relation such that xRy iff x < y. (a) Find RS. (b) Find SR. (c) Illustrate RS and SR via a diagram. (d ) Is the relation (set) RS equal to the relation SR? Why? Solution R = {(1, 3), (3, 5)} S = {(1, 3), (1, 5), (3, 5)}, (a) RS = {(1, 5)} (b) SR = {(1, 5)} AAA RRS 1 1 1 1 (c) 3 3 3 3 5 5 5 5 (d ) ∴ The relations RS = SR Relations are same as the elements of the set. Unit2.indd 34 9/15/2009 3:48:27 PM Relations and Functions • 35 2.8 TYPES OF RELATIONS OR PROPERTIES OF RELATION There are several properties that are used to classify relations on a set. (i) Reflexive: R is reflexive if a R a ∀ a ∈ A. (ii) Symmetric: R is symmetric if a R b ⇒ b R a. (iii) Asymmetric: If R is not symmetric, it is called Asymmetric relation. (iv) Anti-symmetric: R is anti-symmetric if a R b and b R a ⇒ a = b; i.e. if (a, b) ∈ R then (b, a) ∉ R unless a = b (v) Transitive: R is transitive if a R b and b R c ⇒ a R c. ∀ a, b, c. These properties are only defined for a relation on a set. Example 6: If A = {a, b, c, d } and R = {(a, a), (b, b), (c, c), (d, d )}. Prove that R is reflexive. Solution a ∈ A, (a, a) ∈ R, b ∈ A, (b, b) ∈ R, c ∈ A, (c, c) ∈ R, d ∈ A, (d, d ) ∈ R. ∴ a R a ∀ a ∈ A, hence R is reflexive. Example 7: Prove that the relation R = {(1, 1),(2, 1), (2, 2),(2, 3),(2, 4),(3, 1),(3, 2),(3, 3),(3, 4)} on set A = {1, 2, 3, 4} is neither reflexive, nor symmetric or anti-symmetric but transitive. Solution A = {1, 2, 3, 4} (4, 4) ∉ A, hence R is not reflexive. (2, 1) ∈ R and (1, 2) ∉ R, hence R is not symmetric. (2, 3) ∈ R and (3, 2) ∈ R, but 2 ≠ 3, hence R is not anti-symmetric. (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R ∀ a, b, c ∈ A, hence transitive. 2.8.1 Universal Relation A relation R in a set A is called universal relation if R = A × A. e.g.: If A = {x, y, z}. Then R = {(x, x), (x, y), (x, z), ( y, x), ( y, y), ( y, z), (z, x), (z, y), (z, z)} is a universal relation in A. 2.8.2 Identity Relation A relation in a set A is called identity relation if R = {(x, x): x ∈ A} e.g.: If A {x, y, z}. Then R = {(x, x), ( y, y), (z, z)} is an identity relation in A. Unit2.indd 35 9/15/2009 3:48:27 PM 36 • A Textbook of Discrete Mathematics 2.8.3 Equivalence Relation A relation R on a set A is equivalence relation if it is reflexive, symmetric and transitive. Two elements that are related by equivalence relation are called equivalent. 2.8.4 Equivalence Class Equivalence class of an element a ∈A, denoted by [a], having relation R on A, is the set of ele- ments of A to which a is related i.e., [a] = {x: (a, x) ∈ R}. Example 8: Let R be the relation on the set N of positive integers defined by R= {(a, b): a + b is even}. Prove that R is an equivalent relation. Solution R is reflexive:∀ a ∈ N, a + a is even, hence a R a ∀ a. R is symmetric: If a + b is even then b + a is even, hence a R b ⇒ b R a. R is transitive: Let a, b, c have the same parity i.e., these are either odd or even. We see if a R b and b R c then a R c. Hence R is equivalence relation. Example 9: Let R be the relation on the set Z of integers defined by R = {(x, y): x ∈ Z, y ∈ Z, (x - y) is divisible by 6}. Prove that R is an equivalent relation. Solution R is an equivalence relation if it is reflexive, symmetric and transitive. (a) Reflexivity: a - a = 0 is divisible by 6, hence a R a and R is reflexive. (b) Symmetry: a - b = - (b - a), if a - b is divisible by 6, (b - a) is also divisible by 6, hence a R b =b R a and R is symmetric. (c) Transitivity: If a - b and b - c are divisible by 6, let a - b = 6m, m ∈ Z and b - c = 6n, n ∈ Z so that (m + n) ∈Z and (a - c) = 6(m + n), divisible by 6, accord- ingly if a R b and b R c then a R c, so that R is transitive. Example 10: If A is a set of all integers. Then prove that R = {(a, b): a, b ∈ A, (a - b) is an even integer} is an equivalence relation on A. Solution 0 = a – a is an even integer. So (a, a) ∈ R If (a, b) ∈ R, a - b is an even integer.
Recommended publications
  • Relations in Categories

    Relations in Categories

    Relations in Categories Stefan Milius A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of Master of Arts Graduate Program in Mathematics and Statistics York University Toronto, Ontario June 15, 2000 Abstract This thesis investigates relations over a category C relative to an (E; M)-factori- zation system of C. In order to establish the 2-category Rel(C) of relations over C in the first part we discuss sufficient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as defined by Lawvere. We give a significantly simplified proof for the main result of Pavlovi´c,namely that C Map(Rel(C)) if and only if E RegEpi(C). This part also contains a proof' that the category Map(Rel(C))⊆ is finitely complete, and we present the results obtained by Kelly, some of them generalized, i. e., without the restrictive assumption that M Mono(C). The next part deals with factorization⊆ systems in Rel(C). The fact that each set-relation has a canonical image factorization is generalized and shown to yield an (E¯; M¯ )-factorization system in Rel(C) in case M Mono(C). The setting without this condition is studied, as well. We propose a⊆ weaker notion of factorization system for a 2-category, where the commutativity in the universal property of an (E; M)-factorization system is replaced by coherent 2-cells. In the last part certain limits and colimits in Rel(C) are investigated.
  • Simple Laws About Nonprominent Properties of Binary Relations

    Simple Laws About Nonprominent Properties of Binary Relations

    Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation arXiv:1806.05036v2 [math.LO] 20 Nov 2018 Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . 24 4.4 Asymmetry . 24 4.5 Symmetry . 25 4.6 Quasi-transitivity . 26 4.7 Anti-transitivity . 28 4.8 Incomparability-transitivity . 28 4.9 Euclideanness . 33 4.10 Density . 38 4.11 Connex and semi-connex relations . 39 4.12 Seriality . 40 4.13 Uniqueness . 42 4.14 Semi-order property 1 . 43 4.15 Semi-order property 2 . 45 5 Examples 48 6 Implementation issues 62 6.1 Improved relation enumeration . 62 6.2 Quine-McCluskey implementation . 64 6.3 On finding \nice" laws . 66 7 References 69 List of Figures 1 Source code for transitivity check . .5 2 Source code to search for right Euclidean non-transitive relations . .5 3 Timing vs. universe cardinality .
  • What Are Kinship Terminologies, and Why Do We Care?: a Computational Approach To

    What Are Kinship Terminologies, and Why Do We Care?: a Computational Approach To

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Kent Academic Repository What are Kinship Terminologies, and Why do we Care?: A Computational Approach to Analysing Symbolic Domains Dwight Read, UCLA Murray Leaf, University of Texas, Dallas Michael Fischer, University of Kent, Canterbury, Corresponding Author, [email protected] Abstract Kinship is a fundamental feature and basis of human societies. We describe a set of computat ional tools and services, the Kinship Algebra Modeler, and the logic that underlies these. Thes e were developed to improve how we understand both the fundamental facts of kinship, and h ow people use kinship as a resource in their lives. Mathematical formalism applied to cultural concepts is more than an exercise in model building, as it provides a way to represent and exp lore logical consistency and implications. The logic underlying kinship is explored here thro ugh the kin term computations made by users of a terminology when computing the kinship r elation one person has to another by referring to a third person for whom each has a kin term relationship. Kinship Algebra Modeler provides a set of tools, services and an architecture to explore kinship terminologies and their properties in an accessible manner. Keywords Kinship, Algebra, Semantic Domains, Kinship Terminology, Theory Building 1 1. Introduction The Kinship Algebra Modeller (KAM) is a suite of open source software tools and services u nder development to support the elicitation and analysis of kinship terminologies, building al gebraic models of the relations structuring terminologies, and instantiating these models in lar ger contexts to better understand how people pragmatically interpret and employ the logic of kinship relations as a resource in their individual and collective lives.
  • Math 475 Homework #3 March 1, 2010 Section 4.6

    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.
  • Extending Partial Isomorphisms of Finite Graphs

    Extending Partial Isomorphisms of Finite Graphs

    S´eminaire de Master 2`emeann´ee Sous la direction de Thomas Blossier C´edricMilliet — 1er d´ecembre 2004 Extending partial isomorphisms of finite graphs Abstract : This paper aims at proving a theorem given in [4] by Hrushovski concerning finite graphs, and gives a generalization (with- out any proof) of this theorem for a finite structure in a finite relational language. Hrushovski’s therorem is the following : any finite graph G embeds in a finite supergraph H so that any local isomorphism of G extends to an automorphism of H. 1 Introduction In this paper, we call finite graph (G, R) any finite structure G with one binary symmetric reflexive relation R (that is ∀x ∈ G xRx and ∀x, y xRy =⇒ yRx). We call vertex of such a graph any point of G, and edge, any couple (x, y) such that xRy. Geometrically, a finite graph (G, R) is simply a finite set of points, some of them being linked by edges (see picture 1 ). A subgraph (F, R0) of (G, R) is any subset F of G along with the binary relation R0 induced by R on F . H H H J H J H J H J Hqqq HJ Hqqq HJ ¨ ¨ q q ¨ q q q q q Picture 1 — A graph (G, R) and a subgraph (F, R0) of (G, R). We call isomorphism between two graphs (G, R) and (G0,R0) any bijection that preserves the binary relations, that is, any bijection σ that sends an edge on an edge along with σ−1. If (G, R) = (G0,R0), then such a σ is called an automorphism of (G, R).
  • PROBLEM SET THREE: RELATIONS and FUNCTIONS Problem 1

    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.
  • Foundations of Relations and Kleene Algebra

    Foundations of Relations and Kleene Algebra

    Introduction Foundations of Relations and Kleene Algebra Aim: cover the basics about relations and Kleene algebras within the framework of universal algebra Peter Jipsen This is a tutorial Slides give precise definitions, lots of statements Decide which statements are true (can be improved) Chapman University which are false (and perhaps how they can be fixed) [Hint: a list of pages with false statements is at the end] September 4, 2006 Peter Jipsen (Chapman University) Relation algebras and Kleene algebra September 4, 2006 1 / 84 Peter Jipsen (Chapman University) Relation algebras and Kleene algebra September 4, 2006 2 / 84 Prerequisites Algebraic properties of set operation Knowledge of sets, union, intersection, complementation Some basic first-order logic Let U be a set, and P(U)= {X : X ⊆ U} the powerset of U Basic discrete math (e.g. function notation) P(U) is an algebra with operations union ∪, intersection ∩, These notes take an algebraic perspective complementation X − = U \ X Satisfies many identities: e.g. X ∪ Y = Y ∪ X for all X , Y ∈ P(U) Conventions: How can we describe the set of all identities that hold? Minimize distinction between concrete and abstract notation x, y, z, x1,... variables (implicitly universally quantified) Can we decide if a particular identity holds in all powerset algebras? X , Y , Z, X1,... set variables (implicitly universally quantified) These are questions about the equational theory of these algebras f , g, h, f1,... function variables We will consider similar questions about several other types of algebras, a, b, c, a1,... constants in particular relation algebras and Kleene algebras i, j, k, i1,..
  • Mereology Then and Now

    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.
  • Section 4.1 Relations

    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 .
  • LECTURE 1 Relations on a Set 1.1. Cartesian Product and Relations. A

    LECTURE 1 Relations on a Set 1.1. Cartesian Product and Relations. A

    LECTURE 1 Relations on a set PAVEL RU˚ZIˇ CKAˇ Abstract. We define the Cartesian products and the nth Cartesian powers of sets. An n-ary relation on a set is a subset of its nth Carte- sian power. We study the most common properties of binary relations as reflexivity, transitivity and various kinds of symmetries and anti- symmetries. Via these properties we define equivalences, partial orders and pre-orders. Finally we describe the connection between equivalences and partitions of a given set. 1.1. Cartesian product and relations. A Cartesian product M1 ×···× Mn of sets M1,...,Mn is the set of all n-tuples hm1,...,mni satisfying mi ∈ Mi, for all i = {1, 2,...,n}. The Cartesian product of n-copies of a single set M is called an nth-Cartesian power. We denote the nth-Cartesian power of M by M n. In particular, M 1 = M and M 0 is the one-element set {∅}. An n-ary relation on a set M is a subset of M n. Thus unary relations correspond to subsets of M, binary relations to subsets of M 2 = M × M, etc. 1.2. Binary relations. As defined above, a binary relation on a set M is a subset of the Cartesian power M 2 = M × M. Given such a relation, say R ⊂ M × M, we will usually use the notation a R b for ha, bi∈ R, a, b ∈ M. Let us list the some important properties of binary relations. By means of them we define the most common classes of binary relations, namely equivalences, partial orders and quasi-orders.
  • Introduction to Relations

    Introduction to Relations

    CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition:A binary relation from a set A to a set B is a subset R ⊆ A × B: If (a; b) 2 R we say a is related to b by R. A is the domain of R, and B is the codomain of R. If A = B, R is called a binary relation on the set A. Notation: • If (a; b) 2 R, then we write aRb. • If (a; b) 62 R, then we write aR6 b. Discussion Notice that a relation is simply a subset of A × B. If (a; b) 2 R, where R is some relation from A to B, we think of a as being assigned to b. In these senses students often associate relations with functions. In fact, a function is a special case of a relation as you will see in Example 1.2.4. Be warned, however, that a relation may differ from a function in two possible ways. If R is an arbitrary relation from A to B, then • it is possible to have both (a; b) 2 R and (a; b0) 2 R, where b0 6= b; that is, an element in A could be related to any number of elements of B; or • it is possible to have some element a in A that is not related to any element in B at all. 204 1. RELATIONS AND THEIR PROPERTIES 205 Often the relations in our examples do have special properties, but be careful not to assume that a given relation must have any of these properties.
  • The Algebraic Theory of Semigroups the Algebraic Theory of Semigroups

    The Algebraic Theory of Semigroups the Algebraic Theory of Semigroups

    http://dx.doi.org/10.1090/surv/007.1 The Algebraic Theory of Semigroups The Algebraic Theory of Semigroups A. H. Clifford G. B. Preston American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 20-XX. International Standard Serial Number 0076-5376 International Standard Book Number 0-8218-0271-2 Library of Congress Catalog Card Number: 61-15686 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © Copyright 1961 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second Edition, 1964 Reprinted with corrections, 1977. The American Mathematical Society retains all rights except those granted to the United States Government.