1 RELATIONS Learning outcomes Students are able to: a. determine the properties of relations – reflexive, symmetric, transitive, and antisymmetric b. determine.

Slides:



Advertisements
Similar presentations
Representing Relations
Advertisements

Relations Relations on a Set. Properties of Relations.
CSE115/ENGR160 Discrete Mathematics 04/26/12 Ming-Hsuan Yang UC Merced 1.
Basic Properties of Relations
Discrete Mathematics Lecture # 16 Inverse of Relations.
CS2210(22C:19) Discrete Structures Relations Spring 2015 Sukumar Ghosh.
Chapter 7 Relations : the second time around
Discrete Structures Chapter 5 Relations Nurul Amelina Nasharuddin Multimedia Department.
CSE115/ENGR160 Discrete Mathematics 05/03/11 Ming-Hsuan Yang UC Merced 1.
Discrete Structures Chapter 5 Relations and Functions Nurul Amelina Nasharuddin Multimedia Department.
Relations binary relations xRy on sets x  X y  Y R  X  Y Example: “less than” relation from A={0,1,2} to B={1,2,3} use traditional notation 0 < 1,
Discrete Mathematics Lecture#11.
1 Section 7.1 Relations and their properties. 2 Binary relation A binary relation is a set of ordered pairs that expresses a relationship between elements.
Partially Ordered Sets Basic Concepts
Relations Chapter 9.
Chapter 9 1. Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing.
Chapter 9. Chapter Summary Relations and Their Properties Representing Relations Equivalence Relations Partial Orderings.
Chapter 9. Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing Relations.
Discrete Math for CS Binary Relation: A binary relation between sets A and B is a subset of the Cartesian Product A x B. If A = B we say that the relation.
Chapter 9. Section 9.1 Binary Relations Definition: A binary relation R from a set A to a set B is a subset R ⊆ A × B. Example: Let A = { 0, 1,2 } and.
Relations, Functions, and Matrices Mathematical Structures for Computer Science Chapter 4 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Relations, Functions.
Copyright © Cengage Learning. All rights reserved. CHAPTER 8 RELATIONS.
Discrete Mathematics Relation.
Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1.
Relations and their Properties
Relations, Functions, and Matrices Mathematical Structures for Computer Science Chapter 4 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Relations, Functions.
Section 4.4 Properties of Relations. Order Relations Draw an arrow diagram for the relation R defined on the set {1,2,3,4} such that
Discrete Mathematics Lecture # 15 Types of Relations (contd.)
1 Lecture 2 Equivalence Relations Reading: Epp Chp 10.3.
1 Partial Orderings Aaron Bloomfield CS 202 Epp, section ???
Chapter 9. Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing Relations.
Unit II Discrete Structures Relations and Functions SE (Comp.Engg.)
1 Lecture 2: Relations Relations Reading: Epp Chp 10.1, 10.2.
Chapter Relations and Their Properties
1 RELATIONS Learning outcomes Students are able to: a. determine the properties of relations – reflexive, symmetric, transitive, and antisymmetric b. determine.
Chapter 8: Relations. 8.1 Relations and Their Properties Binary relations: Let A and B be any two sets. A binary relation R from A to B, written R : A.
1 Lecture 2 (part 1) Partial Orders Reading: Epp Chp 10.5.
Copyright © Cengage Learning. All rights reserved. CHAPTER 8 RELATIONS.
RelationsCSCE 235, Spring Introduction A relation between elements of two sets is a subset of their Cartesian products (set of all ordered pairs.
Relations and Functions ORDERED PAIRS AND CARTESIAN PRODUCT An ordered pair consists of two elements, say a and b, in which one of them, say a is designated.
Chap. 7 Relations: The Second Time Around
Binary Relations Definition: A binary relation R from a set A to a set B is a subset R ⊆ A × B. Example: Let A = { 0, 1,2 } and B = {a,b} {( 0, a), (
Week 8 - Monday.  What did we talk about last time?  Properties of functions  One-to-one  Onto  Inverses  Cardinality.
1 CMSC 250 Discrete Structures CMSC 250 Lecture 41 May 7, 2008.
Week 8 - Wednesday.  What did we talk about last time?  Relations  Properties of relations  Reflexive  Symmetric  Transitive.
Section 9.1. Section Summary Relations and Functions Properties of Relations Reflexive Relations Symmetric and Antisymmetric Relations Transitive Relations.
Chapter8 Relations 8.1: Relations and their properties.
1 Partial Orderings Epp, section Introduction An equivalence relation is a relation that is reflexive, symmetric, and transitive A partial ordering.
Binary Relation: A binary relation between sets A and B is a subset of the Cartesian Product A x B. If A = B we say that the relation is a relation on.
The Relation Induced by a Partition
Relations and Their Properties
CSE15 Discrete Mathematics 05/03/17
Introduction to Relations
Relations Binary relations represent relationships between the elements of two sets. A binary relation R from set A to set B is defined by: R  A 
Equivalence Relations
Atmiya Institute of Technology & Science
Relations Chapter 9.
Lecture # 14 Types of Relations
Reflexivity, Symmetry, and Transitivity
Aaron Bloomfield CS 202 Epp, section ???
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 9 Relations Slides are adopted from.
CS2210 Discrete Structures Relations
8.5 Equivalence Relations
8.5 Equivalence Relations and 8.6 Partial Ordering
MCS680: Foundations Of Computer Science
Formal Definition and Examples
CSE 321 Discrete Structures
9.5 Equivalence Relations
Lecture # 16 Inverse of Relations
Ø Let the universe set U={1,2,3,...,10}.
Presentation transcript:

1 RELATIONS Learning outcomes Students are able to: a. determine the properties of relations – reflexive, symmetric, transitive, and antisymmetric b. determine equivalence and partial order relations c. represent relations using matrix and graph.

2 Contents Properties of relations Matrix and graph representation of relations Equivalence relations Partial order relations

39 December 2015Relations3 The most basic relation is “=” (e.g. x = y) Generally x R y  TRUE or FALSE –R(x,y) is a more generic representation –R is a binary relation between elements of some set A to some set B, where x  A and y  B

49 December 2015Relations4 Binary relations: xRy On sets x  X y  Y R  X  Y Example: “ less than ” relation from A={0,1,2} to B={1,2,3} Use traditional notation 0 < 1, 0 < 2, 0 < 3, 1 < 2, 1 < 3, 2 < 3 Or use set notation A  B={(0,1),(0,2),(0,3),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)} R={(0,1),(0,2),(0,3), (1,2),(1,3), (2,3)} Or use Arrow Diagrams

59 December 2015Relations5 Formal Definition (Binary) relation from A to B where x  A, y  B, (x,y)  A  B and R  A  B xRy  (x,y)  R Finite example: A={1,2}, B={1,2,3} Infinite example: A = Z (set of integers) and B = Z aRb  a-b  Z even

69 December 2015Relations6 Example Let A = {2,3}, B = {1,3,6} Define a relation R from A to B such that: xRy  x – y is odd How could this explicitly be represented as tuples? –R = {(2,1),(2,3),(3,6)} What if A and B were the set of all integers? –R = {(x,y)  Z  Z |  k  Z such that x – y = 2k + 1}

79 December 2015Relations7 Properties of Relations Reflexive Symmetric Transitive

8 Example Let A = {1,2,3,4}. R 1 ={(1,1),(1,2),(2,2),(2,3),(3,3),(4,4)} R 1 is reflexive. R 2 = {(1,1),(2,2),(3,3)}. R 2 is not reflexive.

9 Example Let A = {1,2,3}. R 1 ={(1,2),(2,1),(1,3),(3,1)} R 1 is symmetric. R 2 = {(1,1),(2,2),(3,3),(2,3)}. R 2 is not symmetric because (3,2) ε R 2.

10 Example (transitive) Let A = {1,2,3,4}. R = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3). R is transitive because (3,2) & (2,1) → ( 3,1) (4,2) & (2,1) → (4,1) (4,3) & (3,1) → (4,1) (4,3) & (3,2) → (4,2)

119 December 2015Relations11 Example Define a relation of A called R –A = {2,3,4,5,6,7,8,9} –R = {(4,4),(4,7),(7,4),(7,7),(2,2),(3,3),(3,6),(3,9), (6,6),(6,3),(6,9),(9,9),(9,3),(9,6)} Draw the arrow diagram Is R  Reflexive? No.  Symmetric? Yes  Transitive? Yes

129 December 2015Relations12 Example A = {0,1,2,3} R over A = {(0,0),(0,1),(0,3),(1,0), (1,1),(2,2),(3,0),(3,3)} Is R –Reflexive? Yes. –Symmetric? Yes. –Transitive? No. (1,0),(0,3) ε R but (1,3) ε R

13 Question ?????

14 Proving Properties on Infinite Sets - “less than” relation Define a relation R on R (the set of all real numbers): For all x, y ε R, x R y ↔ x < y Is R reflexive? symmetric? transitive? R is reflexive iff  x ε R, x R x. By definition of R, this means x < x, for all x ε R. But this is false. Hence, R is not reflexive.

15 “less than” relation R is not symmetric. R is symmetric iff  x,y ε R, if x R y then y R x. By definition of R. this means that  x,y ε R, if x x. But this is false. R is transitive.

169 December 2015Relations16 Congruence Modulo 3 Define a relation R on Z: for all m,n  Z, m R n ↔ 3 | (m – n) R is called congruence modulo 3 Is R reflexive? Is R symmetric? Is R transitive?

17 Properties of Congruence modulo 3 R is reflexive iff for all m in Z, m R m. By definition of R, this means 3 | m – m. or 3|0. This is true since 0 = 0. 3 R is symmetric iff for all m,n in Z, m R n then n R m. By definition of R, this means if 3|(m – n) then 3|(n – m). This is true. m – n = 3k, for some integer k. n – m = - (m – n) = 3(-k). Hence 3|(n – m).

18 Properties of Congruence modulo 3 Is R transitive? It is necessary to show that For all m,n ε Z, if m R n and n R p then m R p. m – n = 3k for some k. n – p = 3l for some l. m – p = (m – n) + (n – p) = 3k + 3l = 3(k + l). Hence 3|(m – p). Therefore, R is transitive.

199 December 2015Relations19 Exercise Define R(x,y) R: Z +  Z + to be {(x,y)  Z +  Z + | x|y} Prove whether or not this is: –Reflexive? –Symmetric? –Transitive?

209 December 2015Relations20 Matrix Representation of a Relation M R = [m ij ] –m ij ={1 iff (i,j)  R and 0 iff (i,j)  R} Example: –R : {1,2,3}  {1,2} R = {(2,1),(3,1),(3,2)}

21 Example

22 Graph Representation of a Relation

239 December 2015Relations23 Union, Intersection, Difference and Composition of relations R: A  B and S: A  B R: A  B and S: B  C

24 Compositions of Relations

259 December 2015Relations25 Example - application Let ID = set of student IDs Let Course = set of courses offered Define relation Summer2007 {(x,y)  ID  Course | student x is registered for course y} Can we do this? Relational databases …

26 Databases ID Course Structure Discrete Java Programming … …

279 December 2015Relations27 Equivalence Relations Any binary relation that is: –Reflexive –Symmetric –Transitive

28 Antisymmetric relation Let R be a relation on a set A. R is antisymmetric iff for all a and b in A, if a R b and b R a then a = b. In other words, a relation is antisymmetric iff there are no pairs of distinct elements a and b with a related to b and b related to a. Let A = {0,1,2} R 1 = {(0,2),(1,2),(2,0)}. R 1 is not antisymmetric R 2 = {(0,0),(0,1),(0,2),(1,1),(1,2)}. R 2 is symmetric.

299 December 2015Relations29 Example Let R 1 be the divides relation on Z + Let R 2 be the divides relation on Z Is R 1 antisymmetric? Prove or give counterexample. –a R 1 b and b R 1 a  a = b –True Is R 2 antisymmetric? Prove or give counterexample. –Counterexample (a = 2, b = -2)

309 December 2015Relations30 Partial Order Relation R is a Partial Order Relation if and only if –R is Reflexive, Antisymmetric and Transitive Partial Order Set (POSET) (S,R) = R is a partial order relation on set S Examples –(Z,  ) –(Z +,|){note: | symbolizes divides} –(S,  ){note: S indicates and set}

31 End Thank you