Chapter 3: Relations and Posets

1. Chapter 3: Relations and Posets Discrete Mathematical Structures: Theory and Applications

2. Learning Objectives • Learn about relations and their basic properties • Explore equivalence relations • Become aware of closures • Learn about posets • Explore how relations are used in the design of relational databases Discrete Mathematical Structures: Theory and Applications

3. Relations • Relations are a natural way to associate objects of various sets Discrete Mathematical Structures: Theory and Applications

4. Relations • R can be described in • Roster form • Set-builder form Discrete Mathematical Structures: Theory and Applications

5. Relations • Arrow Diagram • Write the elements of A in one column • Write the elements B in another column • Draw an arrow from an element, a, of A to an element, b, of B, if (a ,b) R • Here, A = {2,3,5} and B = {7,10,12,30} and R from A into B is definedas follows: For all a  A and b  B, a R b if and only if a divides b • The symbol → (called an arrow) represents the relationR Discrete Mathematical Structures: Theory and Applications

6. Relations Discrete Mathematical Structures: Theory and Applications

7. Relations • Directed Graph • Let R be a relation on a finite set A • Describe Rpictorially as follows: • For each element of A , draw a small or big dot and label the dot by the corresponding element of A • Draw an arrow from a dot labeleda , to another dot labeled, b , ifa R b . • Resulting pictorial representation ofR iscalled the directed graph representation of the relationR Discrete Mathematical Structures: Theory and Applications

8. Relations Discrete Mathematical Structures: Theory and Applications

9. Relations • Directed graph (Digraph) representation of R • Each dot is called a vertex • If a vertex is labeled,a, then it is also called vertexa • An arc from a vertex labeleda, to another vertex,b is called a directed edge, or directed arc froma tob • The ordered pair(A , R) a directed graph, or digraph, of the relationR, where each element of Ais a called a vertex of the digraph Discrete Mathematical Structures: Theory and Applications

10. Relations • Directed graph (Digraph) representation of R (Continued) • For verticesa and b , ifa R b, a is adjacent tob andb is adjacent froma • Because (a, a) R, an arc from a to a is drawn; because (a, b) R, an arc is drawn from a to b. Similarly, arcs are drawn from b to b, b to c , b to a, b to d, and c to d • For an element a A such that (a, a) R, a directed edge is drawn from a to a. Such a directed edge is called a loop at vertex a Discrete Mathematical Structures: Theory and Applications

11. Relations • Directed graph (Digraph) representation of R (Continued) • Position of each vertex is not important • In the digraph of a relationR, there is a directed edge or arc from a vertexa toa vertexb if and only ifa R b • Let A ={a ,b ,c ,d} and let R be the relation defined by the following set: R = {(a ,a ), (a ,b ), (b ,b ), (b ,c ), (b ,a ), (b ,d ), (c ,d )} Discrete Mathematical Structures: Theory and Applications

12. Relations • Domain and Range of the Relation • Let R be a relation from a set A into a set B. Then R ⊆A x B. The elements of the relation R tell which element of A is R-related to which element of B Discrete Mathematical Structures: Theory and Applications

13. Relations Discrete Mathematical Structures: Theory and Applications

14. Relations Discrete Mathematical Structures: Theory and Applications

15. Relations Discrete Mathematical Structures: Theory and Applications

16. Relations • Let A = {1, 2, 3, 4} and B = {p, q, r}. Let R = {(1, q), (2, r ), (3, q), (4, p)}. Then R−1= {(q, 1), (r , 2), (q, 3), (p, 4)} • To find R−1, just reverse the directions of the arrows • D(R) = {1, 2, 3, 4} = Im(R−1), Im(R) = {p, q, r} = D(R−1) Discrete Mathematical Structures: Theory and Applications

17. Relations Discrete Mathematical Structures: Theory and Applications

18. Relations Discrete Mathematical Structures: Theory and Applications

19. Relations • Constructing New Relations from Existing Relations Discrete Mathematical Structures: Theory and Applications

20. Relations Discrete Mathematical Structures: Theory and Applications

21. Relations • Example: • Consider the relations R and S as given in Figure 3.7. • The composition S ◦ R is given by Figure 3.8. Discrete Mathematical Structures: Theory and Applications

22. Relations Discrete Mathematical Structures: Theory and Applications

23. Relations Discrete Mathematical Structures: Theory and Applications

24. Relations Discrete Mathematical Structures: Theory and Applications

25. Relations Discrete Mathematical Structures: Theory and Applications

26. Relations Discrete Mathematical Structures: Theory and Applications

27. Relations Discrete Mathematical Structures: Theory and Applications

28. Relations Example 3.1.26 continued Discrete Mathematical Structures: Theory and Applications

29. Relations Discrete Mathematical Structures: Theory and Applications

30. Relations Example 3.1.27 continued Discrete Mathematical Structures: Theory and Applications

31. Relations Discrete Mathematical Structures: Theory and Applications

32. Relations Example 3.1.31 continued Discrete Mathematical Structures: Theory and Applications

33. Relations Discrete Mathematical Structures: Theory and Applications

34. Relations Example 3.1.32 continued Discrete Mathematical Structures: Theory and Applications

35. Relations Discrete Mathematical Structures: Theory and Applications

36. Relations Discrete Mathematical Structures: Theory and Applications

37. Relations • Consider the relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 4), (4, 1), (2, 3), (3, 2)} on the set S = {1, 2, 3, 4, 5}. • The digraph is divided into three distinct blocks. From these blocks the subsets {1, 4}, {2, 3}, and {5} are formed.These subsets are pairwise disjoint, and their union is S. • A partition of a nonempty set S is a division of S into nonintersecting nonempty subsets. Discrete Mathematical Structures: Theory and Applications

38. Relations Discrete Mathematical Structures: Theory and Applications

39. Relations • Example: Let A denote the set of the lowercase English alphabet. Let B be the set of lowercase consonants and C be the set of lowercase vowels. Then B and C are nonempty, B ∩ C = , and A = B ∪ C. Thus, {B, C} is a partition of A. • Let A be a set and let {A1, A2, A3, A4, A5} be a partition of A. Corresponding to this partition, a Venn diagram, can be drawn, Figure 3.13 Discrete Mathematical Structures: Theory and Applications

40. Relations Discrete Mathematical Structures: Theory and Applications

41. Relations Discrete Mathematical Structures: Theory and Applications

42. Relations Discrete Mathematical Structures: Theory and Applications

43. Relations Discrete Mathematical Structures: Theory and Applications

44. Relations Discrete Mathematical Structures: Theory and Applications

45. Relations • Let A = {a, b, c , d, e , f , g , h, i, j }. Let R be a relation on A such that the digraph of R is as shown in Figure 3.14. • Then a, b, c , d, e , f , c , g is a directed walk in R as a R b,b R c,c R d,d R e, e R f , f R c, c R g. Similarly, a, b, c , g is also a directed walk in R. In the walk a, b, c , d, e , f , c , g , the internal vertices are b, c , d, e , f , and c , which are not distinct as c repeats. • this walk is not a path. In the walk a, b, c , g , the internal vertices are b and c , which are distinct. Therefore, the walk a, b, c, g is a path. Discrete Mathematical Structures: Theory and Applications

46. Relations Discrete Mathematical Structures: Theory and Applications

47. Partially Ordered Sets Discrete Mathematical Structures: Theory and Applications

48. Partially Ordered Sets Discrete Mathematical Structures: Theory and Applications

49. Partially Ordered Sets Discrete Mathematical Structures: Theory and Applications

50. Partially Ordered Sets Discrete Mathematical Structures: Theory and Applications