1. Chapter 8: Relations

2. 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  B, is a subset of the Cartesian product A×B. The notation a R b means (a, b)  R. If a R b we may say that a is related to b (by relation R ), or a relates to b (under relation R ).

3. Example Let R : A  B, and A = {1, 2, 3} represents students, B = {a, b} represents courses. A×B = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}. If R = {(1, a), (1, b)}, it means that student 1 registered in courses a and b

4. Relations can be Represented by: Let A be the set {1, 2, 3, 4} for which ordered pairs are in the relation R = {(a, b) | a divides b} A- Roaster Notation: List of ordered pairs: R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (1, 4), (2, 4)} B- Set builder notation: R = {(a, b) : a divides b}

5. C- Graph: R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (1, 4), (2, 4)} 1 2 3 4 1 2 3 4 OR Relations can be Represented by: Domain of R

6. D- Table: R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (1, 4), (2, 4)} R1234 1×××× 2×× 3× 4× Relations can be Represented by:

7. Relations on a Set A (binary) relation from a set A to itself is called a relation on the set A. e.g. The “ < ” relation defined as a relation on the set N of natural numbers: let < : N  N :≡ {(a, b) | a < b } If (a, b)  R then a < b means (a, b)  < e.g. (1, 2)  <.

8. Examples: Consider the relations on A = {-1, 0, 1, 2}, R 1 = {(a, b) | a < b} R 2 = {(a, b) | a = b or a = -b} R 3 = {(a, b) | 0 ≤ a + b ≤ 1} R 1 = {(-1,0), (-1,1), (-1,2), (0,1), (0,2), (1,2)} R 2 = {(-1,-1), (0,0), (1,1), (2,2), (-1,1), (1,-1)} R 3 = {(-1,1), (1,-1), (-1,2), (2,-1), (0,0), (0,1), (1,0)} Relations on a Set

9. The identity relation I A on a set A is the set {(a, a) | a  A}. e.g. If A = {1, 2, 3, 4}, then I A = {(1, 1), (2, 2), (3, 3), (4, 4)}.

10. Question How many relations are there on a set with n elements? Answer: 1. A relation on set A is a subset from A×A. 2. A has n elements so A×A has n 2 elements. 3. Number of subsets for n 2 elements is, thus there are relations on a set with n elements. e.g. If S = {a, b, c}, there are relations.

11. Properties of Relations 1. Reflexivity and Irreflexivity A relation R on A is reflexive if (a, a)  R for every element a  A. e.g. Consider the following relations on {1, 2, 3, 4} R 1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}, Not Reflexive. R 2 = {(1, 1), (2, 1), (2, 2), (3, 3), (3, 4), (4, 4)}, Reflexive. R 3 = {(a, b) | a ≤ b}, Reflexive.

12. A relation R on A is irreflexive if for every element a  A, (a, a)  R. Note: “irreflexive” ≠ “not reflexive”. e.g. If A = {1, 2}, R = {(1, 2), (2, 1), (1, 1)} is not reflexive because (2, 2)  R, not irrflexive because (1, 1)  R. Reflexivity and Irreflexivity

13. 1 2 3 4 Not Reflexive and Not Irreflexive Example

14. 1 2 1 2 3 4 3 4 Irreflexive Reflexive Examples

15. 2. Symmetry and Antisymmetry A binary relation R on A is symmetric if (a, b)  R ↔ (b, a)  R, where a, b  A. A binary relation R on A is antisymmetric if (a, b)  R → (b, a)  R. That is, if (a, b)  R  (b, a)  R → a = b.

16. Examples Consider these relations on the set of integers: R 1 = {(a, b) | a = b} Symmetric, antisymmetric. R 2 = {(a, b) | a > b}, Not symmetric, antisymmetric. R 3 = {(a, b) | a = b + 1}, Not symmetric, antisymmetric.

17. Examples Let A = {1, 2, 3}. R 1 = {(1, 2), (2, 2), (3, 1), (1, 3)} Not reflexive, not irreflexive, not symmetric, not antisymmetric R 2 = {(2, 2), (1, 3), (3, 2)} Not reflexive, not irreflexive, not symmetric, antisymmetric R 3 = {(1, 1), (2, 2), (3, 3)} Reflexive, not irreflexive, symmetric, antisymmetric R 4 = {(2, 3)} Not reflexive, irreflexive, not symmetric, antisymmetric

18. 3. Transitivity A relation R is said to be transitive if and only if (for all a, b, c), (a, b)  R  (b, c)  R → (a, c)  R. e.g. Let A = {1, 2}. R 1 = {(1, 1), (1, 2), (2, 1), (2, 2)} is transitive. R 2 = {(1, 1), (1, 2), (2, 1)} is not transitive, (2, 2)  R 2. R 3 = {(3, 4)} is transitive.

19. Special Cases Empty set { } Irreflexive, transitive, symmetric, antisymmetric. Universal set U Reflexive, transitive, symmetric.

20. Combining Relations Let A = {1, 2, 3}, B = {1, 2, 3, 4}, R 1 = {(1, 1), (2, 2), (3, 3)}, R 2 = {(1, 1), (1, 2), (1, 3), (1, 4)}, then R 1 R 2 = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (1, 4)} R 1 R 2 = {(1, 1)} R 1 − R 2 = {(2, 2), (3, 3)}

21. Composite Relations If (a, c) is in R 1 and (c, b) is in R 2 then (a, b) is in R 2 ◦R 1. e.g. R is the relation from {1, 2, 3} to {1, 2, 3, 4} R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)}. S is the relation from {1, 2, 3, 4} to {0, 1, 2} S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}. S◦R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)}.

22. Powers Let R be a relation on the set A. the power R n, n = 1, 2, 3, … are defined by R 1 = R and R n = R n -1 ◦ R e.g. Let R = {(1, 1), (2, 1), (3, 2), (4, 3)}. Find R 2 =R ◦ R = {(1, 1), (2, 1), (3, 1), (4, 2)} R 3 = R 2 ◦ R = {(1, 1), (2, 1), (3, 1), (4, 1)}

23. 8.3 Representing Relations Some special ways to represent binary relations: –With a zero-one matrix. –With a directed graph.

24. Using Zero-One Matrices To represent a relation R by a matrix M R = [m ij ], let m ij = 1 if (a i, b j )  R, otherwise 0. e.g., Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally. The 0-1 matrix representation of that relation:

25. Let A = {1, 2, 3}, B = {1, 2}, R : A → B such that: R = {(2, 1), (3, 1), (3, 2)} then the matrix for R is: 1 2 1 2 3 Example

26. Zero-One Reflexive, Symmetric The terms: Reflexive, non-reflexive, irreflexive, symmetric and antisymmetric. –These relation characteristics are very easy to recognize by inspection of the zero-one matrix. Reflexive: all 1’s on diagonal Irreflexive: all 0’s on diagonal Symmetric: all identical across diagonal Antisymmetric: all 1’s are across from 0’s any- thing anything

27. Example Is R reflexive, symmetric, antisymmetric? Reflexive, symmetric, not antisymmetric

28. Operations 1- Union and the Intersection The Boolean Operations join and meet can be used to find the matrices representing the union and the intersection of two relations

29. Example Suppose R 1 and R 2 are relations on a set A which are represented by the matrices:

30. 2- Composite Suppose that R : A ↔ B, S : B ↔ C (Boolean Product) Operations

31. Example Let Find the matrix of ?

32. 3- Power Operations

33. Example Find the matrix that represents R 2 where the matrix representing R is:

34. Using Directed Graphs A directed graph or digraph G = (V G, E G ) is a set V G of vertices (nodes) with a set E G  V G ×V G of edges (arcs or links). Visually represented using dots for nodes, and arrows for edges. Notice that a relation R : A ↔ B can be represented as a graph G R = (V G = A  B, E G = R). MRMR GRGR Joe Fred Mark Susan Mary Sally Node set V G (black dots) Edge set E G (blue arrows)

35. Digraph Reflexive, Symmetric It is extremely easy to recognize the reflexive, irreflexive, symmetric, antisymmetric properties by graph inspection.       Reflexive: Every node has a self-loop Irreflexive: No node links to itself Symmetric: Every link is bidirectional   Antisymmetric: No link is bidirectional    Not symmetric, non-antisymmetricNon-reflexive, non-irreflexive

36. 8.4 Closures of Relations For any property X, the “X closure” of a set A is defined as the “smallest” superset of A that has the given property. The reflexive closure of a relation R on A is obtained by adding (a, a) to R for each a  A not already in R. i.e. It is R  I A.

37. Example 1 The relation R = {(1, 1), (1, 2), (2, 1), (3, 2)} on the set A = {1, 2, 3} is not reflexive. How can you produce a reflexive relation containing R that is as small as possible? Answer: By adding (2, 2) and (3, 3) so the reflexive closure of R is: {(1, 1), (1, 2), (2, 1), (3, 2), (2, 2), (3, 3)}.

38. Example 2 What is the reflexive closure of the relation: R = {(a,b) | a < b} on the set of integers? Answer: The reflexive closure of R is: {(a, b) | a < b} {(a, a) | a  Z} = {(a, b) | a ≤ b}.

39. Symmetric Closure The symmetric closure of R is obtained by adding (b, a ) to R for each (a, b ) in R. i.e. It is R  R −1.

40. Example 1 The relation {(1, 1), (2, 2), (1, 2), (3, 1), (2, 3), (3,2)} on the set {1, 2, 3} is not symmetric. How can we produce a symmetric relation that is as small as possible and contains R ? Answer: By adding (2, 1) and (1, 3) so the symmetric closure of R is: {(1, 1), (2, 2), (1, 2), (3, 1), (2, 3), (3, 2), (2, 1), (1, 3)}.

41. Example 2 What is the symmetric closure of the relation: R = {(a, b) | a > b} on the set of positive integers? Answer: The Symmetric Closure of R is: {(a, b) | a > b} {(b, a) | a > b} = {(a, b) | a ≠ b}.

42. Transitive Closure The transitive closure or connectivity relation of R is obtained by repeatedly adding (a, c) to R for each (a, b), (b, c) in R. –i.e. It is – Or in term of zero-one matrices:

43. Example 1 e.g. R = {(1,1), (1, 2), (2,1), (3, 2)} on the set A = {1, 2, 3} R* = R R 2 R 3 R 2 = R o R = {(1,1), (1, 2), (2, 2), (3,1)} R 3 = R 2 o R = {(1,1), (1, 2), (2,1), (2, 2), (3, 2)} R* = {(1,1), (1, 2), (2,1), (3, 2), (2, 2), (3,1)}

44. Example 2 Find M R* for

45. 8.5 Equivalence Relations An equivalence relation on a set A is simply any binary relation on A that is reflexive, symmetric, and transitive. Example1: Show that a relation on the set of real numbers R = {(a, b) | a – b is an integer}. is an equivalence relation. R is reflexive: Since a – a = 0 is an integer  a R a for all a. R is symmetric: If a R b then whenever a – b is an integer  b – a is an integer  b R a. R is transitive: If a R b and b R c then a – b and b – c are integers, therefore, a – b + b – c = a – c is also an integer, which means a R c.

46. Example 2: Let R : Z  Z. Show that R = {(a, b) | a ≡ b (mod m)}, m > 1 is an equivalence relation. Answer: a ≡ b (mod m) if and only if m divides a – b R is reflexive, a ≡ a (mod m), m divides a – a = 0. R is symmetric: Suppose a ≡ b (mod m) then m divides a – b, therefore m divides –(a – b) = b – a  b ≡ a (mod m). R is transitive: Suppose a ≡ b (mod m) and b ≡ c (mod m), then: m divides both a – b and b – c, therefore, m divides their sum (a – b) + (b – c) = a – c, thus a ≡ c (mod m). Equivalence Relations

47. Example 3: Let R be a relation on Z defined as follows: a R b if and only if 5 | (a 2 – b 2 ). Show that R is an equivalence relation. R is reflexive: a 2 – a 2 = 0 = 0×5  5 | (a 2 – a 2 )  a R a. R is symmetric: If a R b  5 | (a 2 – b 2 )  a 2 – b 2 = m×5, m  Z  b 2 – a 2 = (-m)×5  5 | (b 2 – a 2 )  b R a. R is transitive: Suppose that a R b  5 | (a 2 – b 2 )  a 2 – b 2 = m×5, m  Z and b R c  5 | (b 2 – c 2 )  b 2 – c 2 = k×5, k  Z. Now (a 2 – b 2 ) + (b 2 – c 2 ) = (a 2 – c 2 ) =(m + k)×5  5 | (a 2 – c 2 )  a R c. Equivalence Relations

48. Equivalence Classes Let R be an equivalence relation on a set A. The set of all elements that are related to an element a of A is called the equivalence class of a. The equivalence class of a with respect to R is denoted by [a] R. When only one relation is under consideration, we can delete the subscript R and write [a] for this equivalence class. [a] R = {s | (a, s)  R }.

49. Example: What are the equivalence classes of 0 and 1 for congruence modulo 4? Solution: The equivalence class of 0 contains all integers a such that a ≡ 0 (mod 4). The integers in this class are those divisible by 4. Hence, the equivalence class of 0 for this relation is [0] = {..., -8, -4, 0, 4, 8,... }. The equivalence class of 1 contains all the integers a such that a ≡ 1 (mod 4). The integers in this class are those that have a remainder of 1 when divided by 4. Hence, the equivalence class of 1 for this relation is [1] = {..., -7, -3, 1, 5, 9,... }. Equivalence Classes

50. Equivalence Classes and Partitions A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. In other words, the collection of subsets A i, i  I (where I is an index set) forms a partition of S if and only if

51. Example: What are the sets in the partition of the integers arising from congruence modulo 4? Solution: There are four congruence classes, corresponding to [0] 4, [1] 4, [2] 4, and [3] 4. They are the sets [0] 4 = {..., -8, -4, 0, 4, 8,... }, [1] 4 = {..., -7, -3, 1, 5, 9,... }, [2] 4 = {..., -6, -2, 2, 6, 10,... }, [3] 4 = {..., -5, -1, 3, 7, 11,... }. These congruence classes are disjoint, and every integer is in exactly one of them. In other words, these congruence classes form a partition. Equivalence Classes and Partitions

52. 8.6 Partial Orderings A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R). Members of S are called elements of the poset.

53. Example 1 Show that the "greater than or equal" relation ( ) is a partial ordering on the set of integers. Solution: a ≥ a for every integer a, ≥ is reflexive. If a ≥ b and b ≥ a, then a = b. Hence, ≥ is antisymmetric. Finally, ≥ is transitive because a ≥ b and b ≥ c imply that a ≥ c. It follows that ≥ is a partial ordering on the set of integers and (Z, ≥) is a poset.

54. Example 2 Show that the divisibility relation “|“ is a partial ordering on the set of positive integers Z +. Solution: a | a for every positive integer a, | is reflexive. If a | b and b | a, then a = b, | is antisymmetric. Finally, | is transitive because a | b and b | c imply that a | c. We see that (Z +, |) is a poset.

55. The elements a and b of a poset (S, ) are called comparable if either a b or b a. When a and b are elements of S such that neither a b nor b a, a and b are called incomparable. Example: In the poset (Z +, |), The integers 3 and 9 are comparable because 3 | 9. The integers 5 and 7 are incomparable, because 5 | 7 and 7 | 5. Definition

56. If (S, ) is a poset and every two elements of S are comparable, S is called a totally ordered or linearly ordered set, and is called a total order or a linear order. A totally ordered set is also called a chain. Example 1: The poset (Z, ≤) is totally ordered, because a ≤ b or b ≤ a whenever a and b are integers. Example 2: The poset (Z +, |) is not totally ordered because it contains elements that are incomparable, such as 5 and 7. Definition

57. Hasse Diagrams We can represent a partial ordering on a finite set using this procedure: Start with the directed graph for this relation. Because a partial ordering is reflexive, a loop is present at every vertex. Remove these loops. Remove all edges that must be in the partial ordering because of the presence of other edges and transitivity. For instance, if (a, b) and (b, c) are in the partial ordering, remove the edge (a, c), because it must be present also. Furthermore, if (c, d) is also in the partial ordering, remove the edge (a, d), because it must be present also. Finally, arrange each edge so that its initial vertex is below its terminal vertex (as it is drawn on paper). Remove all the arrows on the directed edges, because all edges point "upward" toward their terminal vertex. (The edges left correspond to pairs in the covering relation of the poset.

58. Example 1 Constructing the Hasse Diagram for ({ 1, 2, 3, 4 }, ≤).

59. Example 1 Let A = {a, b, c}. Constructing the Hasse Diagram for (P(A),  ). P(A) = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}  {a}{a} {c}{c} {b}{b} {a, b} {a, c} {b, c} {a, b, c}

60. Maximal and Minimal Elements Elements of posets that have certain extremal properties are important for many applications. An element of a poset is called maximal if it is not less than any element of the poset. That is, a is maximal in the poset (S, ) if there is no b S such that a b. Similarly, an element of a poset is called minimal if it is not greater than any element of the poset. That is, a is minimal if there is no element b S such that b a. Maximal and minimal elements are easy to spot using a Hasse diagram. They are the "top" and "bottom" elements in the diagram.

61. Which elements of the Poset ({2, 4, 5,10, 12, 20, 25 }, | ) are maximal, and which are minimal? Solution: Maximal elements are 12, 20 and 25. Minimal elements are 2 and 5. The Hasse Diagram of the Poset. This example shows that a poset can have more than one maximal element and more than one minimal element. Example

62. Greatest Element and Least Element We say that a is the greatest element of the poset (S, ) if b a for all b  S. The greatest element is unique when it exists. Also, a is siad the least element of the poset (S, ) if a b for all b  S. The least element is unique when it exists.

63. Determine whether the posets represented by each of the Hasse diagrams in the Figure have a greatest element and/or a least element. Answer: (a) No greatest element, a is the least element. (b) No greatest element, no least element. (c) d is the greatest element, no least element. (d) d is the greatest element, a is the least element. Example

64. If A is a subset of the poset (S, ) and u is an element of S such that a u for all elements a A, then u is called an upper bound of A. The element x is called the least upper bound of the subset A if x is an upper bound that is less than every other upper bound of A. If l is an element of S such that I a for all elements a A, then I is called a lower bound of A. The element y is called greatest lower bound of A if y is a lower bound of A and z y whenever z is a lower bound of A. The greatest lower bound and least upper bound of a subset A are denoted by glb(A) and lub(A), respectively. Example

65. Consider the poset ({2, 4, 5,10, 12, 20, 25 }, | ). Find the lower and upper bounds of A = {2, 4}. Find the glb(A) and lub(A). Solution: The lower bound of A is 2. The upper bound of A is 4,12 and 20. The glb(A) is 2. The lub(A) is 4. The Hasse Diagram of the Poset. Example