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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 . -
Binary Relations.Pdf
Binary Relations Any set of ordered pairs defines a binary relation. We express a particular ordered pair, (x, y) R, where R is a binary relation, as xRy. Properties Properties of a binary relation R on a set X: a. reflexive: if for every x X, xRx holds, i.e. (x, x) R. b. symmetric: if for every x and y in X, whenever xRy holds, then yRx holds. c. transitive: if for every x, y, and z in X, whenever xRy holds and yRz holds, then xRz holds. Graphical Representation Suppose that X = {A, B, C, D} and that R = {(A, B), (B, D), (C, C), (D, B)}. We can represent the binary relation R as a set, as a graph or as a Boolean matrix. Graphically, we have Boolean Matrix Representation As a Boolean matrix we have A B C D A 0 1 0 0 B 0 0 0 1 C 0 0 1 0 D 0 1 0 0 What would a Boolean matrix look like if it represented a reflexive binary relation, a symmetric binary relation, a transitive binary relation, any combination of the above? Reflexive Relation A B C D A - 1 - - B - - - 1 C - - 1 - D - 1 - - Symmetric Relation A B C D A - 1 - - B - - - 1 C - - 1 - D - 1 - - Transitive Relation A B C D A - 1 - - B - - - 1 C - - 1 - D - 1 - - Click here for Answers Composition of Relations: Let R be a relation from X to Y, and S be a relation from Y to Z. Then a relation written as R o S is called a composite relation of R and S where R o S = { (x, z) | x X z Z ( y) (y Y (x, y) R (y, z) S) } We can also write the composition as R o S = { (x, z) | x X z Z ( y) (y Y xRy ySz) } Note: Relational composition can be realized as matrix multiplication. -
Packet #5: Binary Relations Applied Discrete Mathematics
CSC224/226 – Packet 5 – Binary Relations Packet #5: Binary Relations Applied Discrete Mathematics Table of Contents Binary Relations Summary Page 1 Binary Relations Examples Page 2 Properties of Relations Page 3 Examples Pages 4-5 Representations Page 6 Composites Page 8 Closures Pages 10-11 Equivalence Relations Pages 12-13 Posets & Hasse Diagrams Pages 14-17 CSC224 – Packet 5 – Binary Relations CSC224/226 – Packet 5 – Binary Relations Binary Relations Summary I. Definition: A binary relation is a set of points, or ordered pairs. The ordered pairs are of the form (a,b) where a is a member of a set A (a ∈ A) and b is a member of a set B (b ∈ B). Any subset R ⊆ A x B is a binary relation. (Remember that A x B is the set of ordered pairs {(a,b) | a∈A ∧ b ∈ B}) II. Properties: Consider the binary relation R ⊆ A x A. A. Reflexive:R is reflexive if ∀ a ∈ A, (a,a) ∈ R B. Irreflexive: R is irreflexive if ∀ a ∈ A, (a,a) ∉ R. Is ¬ (R is reflexive) ⇔ (R is irreflexive)? C. Non-reflexive: ¬ (R is reflexive) D. Symmetric: R is symmetric if ∀ a, b ∈ A, [ ((a,b) ∈ R) ⇔ ((b,a) ∈ R) ] E. Antisymmetric: R is antisymmetric if ∀ a, b ∈ A, [(a,b) ∈ R ∧ (b,a) ∈ R] ⇒ (b = a) F. Asymmetric: R is asymmetric if ∀ a, b ∈ A, [(a,b) ∈ R] ⇒ [(b,a) ∉ R] Asymmetric = irreflexive AND antisymmetric G. Transitive: R is transitive if ∀ a, b ∈ A, [(a,b) ∈ R ∧ (b,c) ∈ R] ⇒ (a,c) ∈ R Also: R2 ⊆ R if and only if R is transitive. -
Chapter 5 Partial Orders, Lattices, Well Founded Orderings
Chapter 5 Partial Orders, Lattices, Well Founded Orderings, Equivalence Relations, Distributive Lattices, Boolean Algebras, Heyting Algebras 5.1 Partial Orders There are two main kinds of relations that play a very important role in mathematics and computer science: 1. Partial orders 2. Equivalence relations. In this section and the next few ones, we define partial orders and investigate some of their properties. 459 460 CHAPTER 5. PARTIAL ORDERS, EQUIVALENCE RELATIONS, LATTICES As we will see, the ability to use induction is intimately related to a very special property of partial orders known as well-foundedness. Intuitively, the notion of order among elements of a set, X, captures the fact some elements are bigger than oth- ers, perhaps more important, or perhaps that they carry more information. For example, we are all familiar with the natural ordering, , of the integers ≤ , 3 2 1 0 1 2 3 , ··· − ≤− ≤− ≤ ≤ ≤ ≤ ≤··· the ordering of the rationals (where p1 p2 iff q1 q2 p2q1 p1q2 ≤ − 0, i.e., p2q1 p1q2 0ifq1q2 > 0 else q1q2 ≥ − ≥ p2q1 p1q2 0ifq1q2 < 0), and the ordering of the real numbers.− ≤ In all of the above orderings, note that for any two number a and b, either a b or b a. ≤ ≤ We say that such orderings are total orderings. 5.1. PARTIAL ORDERS 461 A natural example of an ordering which is not total is provided by the subset ordering. Given a set, X, we can order the subsets of X by the subset relation: A B, where A, B are any subsets of X. ⊆ For example, if X = a, b, c , we have a a, b . -
Relations III
CS 441 Discrete Mathematics for CS Lecture 23 Relations III. Milos Hauskrecht [email protected] 5329 Sennott Square CS 441 Discrete mathematics for CS M. Hauskrecht Composite of relations Definition: Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of the ordered pairs (a,c) where a A and c C, and for which there is a b B such that (a,b) R and (b,c) S. We denote the composite of R and S by S o R. Example: AB C c a b RS CS 441 Discrete mathematics for CS M. Hauskrecht 1 Composite of relations Definition: Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of the ordered pairs (a,c) where a A and c C, and for which there is a b B such that (a,b) R and (b,c) S. We denote the composite of R and S by S o R. (a,c) S o R Example: AB C c a b RS CS 441 Discrete mathematics for CS M. Hauskrecht Composite of relations Definition: Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of the ordered pairs (a,c) where a A and c C, and for which there is a b B such that (a,b) R and (b,c) S. -
CS202 Fall 2011 Lecture (Lost Count) – 12/1 Relations
11/30/11 Relations CS202 Fall 2011 Lecture (lost count) – 12/1 Recall the definition of the Cartesian (Cross) Product: The Cartesian Product of sets A and B, A x B, is the set Relations A x B = {(a,b) : a∈A and b∈B}. Prof. Tanya Berger-Wolf aRb or (a,b) ∈ R means “a is related to b” A relation is just any subset of the Cartesian Product: R ⊆ AxB Ex1: A = {0,1,2}, B = {2,3} => AxB = {(0,2), (0,3), (1,2), (1,3), (2,2), (2,3)} R = {(a,b) | a < b}. So R = {(0,2), (0,3), (1,2), (1,3), (2,3)} = AxB-{(2,2)} Ex2: A = students at UIC; B = courses at UIC. R = {(a,b) | student a is enrolled in class b} Ex3: A = {3 letter strings}, B = {all English words} R = {(a,b) | a is a prefix of b} 1 Relations and Functions Properties of Relations Recall the definition of a function: f = {(a,b) : b = f(a) , a∈A and b∈B} Is every function a relation? Yes, a function is Reflexivity: a special kind of A relation R on AxA is reflexive if for all a∈A, (a,a) ∈R. relation. Symmetry: Draw Venn diagram of cross products, relations, Cross product A relation R on AxA is symmetric if functions (a,b) ∈ R implies (b,a) ∈ R . 1 11/30/11 Properties of Relations Properties of Relations - techniques… How can we check for transitivity? Draw a picture of the relation (called a “graph”) [Epp p. 580]. Transitivity: " ! Vertex for every element of A A relation on AxA is transitive if " ! Edge for every element of R (a,b) ∈ R and (b,c) ∈ R imply (a,c) ∈ R. -
Today's Lecture Chapter 8: Section
Announcements ICS 6B } Regrades for Quiz #3 and Homeworks #4 & Boolean Algebra & Logic 5 are due Thursday Lecture Notes for Summer Quarter, 2008 Michele Rousseau Set 7 – Ch. 8.4, 8.5, 8.6 Lecture Set 7 - Chpts 8.4, 8.5, 8.6 2 Grades Overall Grades & Quiz #3… Quiz #3 90-100 } Max: 100% } Min: 42% 80-89 } Median: 83% 70-79 Overall 60-69 less } Max: 99% than 50 } Min 45% 01234567 } Median: 84% Overall Quiz #3 Lecture Set 7 - Chpts 8.4, 8.5, 8.6 3 Lecture Set 7 - Chpts 8.4, 8.5, 8.6 4 Today’s Lecture } Chapter 8 8.4, 8.5, 8.6 ● Closures of Relations 8.4 Chapter 8: Section 8.4 ● Equivalence Relations 8.5 ● Partial Orderings 8.6 Closures of Relations Lecture Set 7 - Chpts 8.4, 8.5, 8.6 5 1 Closure of Relations Reflexive Closure Let R be a relation on set A. Example Let P be a property (reflexive, symmetric, etc.) A1,2,3 The closure of R with respect to the property P R1,1,1,2,1,3 is the smallest relation containing R which has this property. P“being reflexive” } In other words, add the minimum number of pairs to obtain R is not reflexive, b/c its missing 2,2, 3,3 property P. The smallest reflexive relation containing R is } Note: This may not be possible. S1,1,1,2,1,3,2,2,3,3 Example: This is the reflexive closure of R & it’s the intersection A1,2,3,4, R1,1,1,3,14 of all of the reflexive relations that contain R P is being “irreflexive” Any relation on A which is reflexive and contains R } If the closure S of R w.r.t. -
Relations and Functions
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.