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8.3 Representing Relations

8.3 Representing Relations. Directed Graphs Vertex Arc (directed edge) Initial vertex Terminal vertex. Example. Draw the “divides” relation on the set {2,3,4,5,6,7,8,9} as a directed graph. The zero-one Matrix Representation M R.

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8.3 Representing Relations

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  1. 8.3 Representing Relations • Directed Graphs • Vertex • Arc (directed edge) • Initial vertex • Terminal vertex

  2. Example • Draw the “divides” relation on the set {2,3,4,5,6,7,8,9} as a directed graph

  3. The zero-one Matrix Representation MR • MR is just a zero-one version of the “chart” representation of R.

  4. Reflexivity Directed graph picture Zero-one matrix picture

  5. Symmetry Directed graph picture Zero-one matrix picture

  6. Antisymmetry Directed graph picture Zero-one matrix picture

  7. Transitivity Directed graph picture Zero-one matrix picture

  8. Theorem For relations R1 and R2 on set A,

  9. Example: Let and be binary relations. Find and . Use them to find Verify by calculating without matrices.

  10. Corollary For a relation R on set A, for any positive integer n.

  11. Example: Let and . Calculate and to determine if the relations and are transitive.

  12. 8.4 Closures of Relations • Reflexive closure • Symmetric closure

  13. Example: Let R be the relation on the set containing the pairs What is the reflexive closure of R? What is the symmetric closure of R?

  14. Paths in Directed Graphs • A path in a directed graph is a sequence of vertices for which any two consecutive vertices ai and ai+1 in the sequence are joined by an arc from ai to ai+1. • Theorem: Let R be a relation on set A, and n a positive integer. Then there is a path of length n from a to b in R if and only if (a,b) is in Rn.

  15. Example:

  16. The “Connectivity Relation” R* • Let R be a relation on set A. We define

  17. Example: Let be the relation on the set of all people in the world that contains if has met . What is , where is a positive integer greater than one? What is ?

  18. The Transitive Closure • For a relation R on a set A, we define the transitive closure of R to be the smallest transitive relation containing R. • Theorem:

  19. Finding transitive closure the “hard” way:

  20. Computing R* • If A is a set with n elements, and R is a relation on A, then any time there is a path of length one or more from a to b in R then there is a path of length n or less. • So actually and • Interestingly, this is not the best way of computing R*

  21. Computing transitive closure the better way:

  22. Warshall’s Algorithm procedure Warshall(MR: n by n zero-one matrix) W := MR for k:=1 to n for i:=1 to n for j:=1 to n wij := wij (wik  wkj) { W now contains MR* }

  23. Illustration of Warshall’s Algorithm

  24. 8.5 Equivalence Relations • Definition: A relation R on set A is an equivalence relation if …

  25. Examples • aRb if and only if a and b have the same first name (on the set of students in this class) • aRb if and only if a ≡ b (mod 5) (on the set of integers)

  26. Equivalence Classes • If R is a relation on set A, and a is an element of A, then…

  27. Examples (continued) • [Michael] • [4]5

  28. Theorem For an equivalence relation R on set A and elements a and b of A, the following are all logically equivalent: • a R b • [a]R = [b]R • [a]R [b]R  

  29. Partitions • For a set S, a partition of S is a collection  = {A1, A2, …, Am} of nonempty subsets of S which satisfies the following properties: • Every element of A is in one of the sets Ai. • For all i, j  {1, 2, …, m}, if i  j then Ai Aj =  • Terminology: We say that the collection partitions S.

  30. Theorem • Let R be an equivalence relation on set S. Then the equivalence classes of R partition S. Conversely, for any partition  of S there is an equivalence relation R whose equivalence classes are the sets in .

  31. …,-10, , ,-7, , ,-4, , ,-1, , ,2, , ,5, , ,8, , , … …, ,-9, , ,-6, , ,-3, , ,0, , ,3, , ,6, , ,9, , … …, , ,-8, , ,-5, , ,-2, , ,1, , ,4, , ,7, , ,10, … Visual …, ,-9, , ,-6, , ,-3, , ,0, , ,3, , ,6, , ,9, , … …, , ,-8, , ,-5, , ,-2, , ,1, , ,4, , ,7, , ,10, … …,-10, , ,-7, , ,-4, , ,-1, , ,2, , ,5, , ,8, , , … [2]={… ,-10, -7, -4, -1, 2, 5, 8, …} [0]={… ,-9, -6, -3, 0, 3, 6, 9, …} [1]={… ,-8, -5, -2, 1, 4, 7, 10, …}

  32. Visual [2]={… ,-10, -7, -4, -1, 2, 5, 8, …} [0]={… ,-9, -6, -3, 0, 3, 6, 9, …} [1]={… ,-8, -5, -2, 1, 4, 7, 10, …} …,-10, , ,-7, , ,-4, , ,-1, , ,2, , ,5, , ,8, , , … …, ,-9, , ,-6, , ,-3, , ,0, , ,3, , ,6, , ,9, , … …, , ,-8, , ,-5, , ,-2, , ,1, , ,4, , ,7, , ,10, …

  33. Example: Turning a partition into an equivalence relation

  34. Zero-One Matrix Representation of an Equivalence Relation Examples

  35. Digraph Representation of an Equivalence Relation Examples

  36. Equivalence as “sameness” • Almost every equivalence relation definition comes down to identifying some notion of “sameness” • Same remainder when divided by n • Same name • Same set of a partition

  37. Number of Partitions of a Set with n Elements • n = 1 • n = 2 • n = 3 • n = 4

  38. Recurrence Relation for the Number of Partitions of a Set with n Elements

  39. 8.6 Partial Orderings Let A be a set, and R a relation on A. We say that R is a partial ordering if and only if… In this case we say that the pair (A, R) is a partially ordered set (poset).

  40. Examples: • The real numbers R under the  relation • The real numbers R under the  relation • The positive integers under the “divides” relation • Any set of sets under the  (subset) relation • The cartesian product ZZunder the “(,)” relation R. (i.e. (x,y) R (z,w) if and only if x  z and y  w.)

  41. Convention • The symbol is the default symbol used to represent a partial ordering. • Example: “Let A be a set, and let be a partial ordering on A.”

  42. Comparable and Incomparable Elements • Two elements aand bof a partially ordered set are said to be incomparable if and only if the statements ab and ba are both false. Otherwise the elements are comparable. • Examples: • Subsets • Cartesian products • Divides relation

  43. Examples:

  44. Total Orderings • Let A be a set, and let be a partial ordering on A. We say that is a total ordering provided…In this case we say that the pair (A, ) is a totally ordered set. (linearly ordered set, chain) • Examples: 1) Real numbers under  2) Any set of strings under the “dictionary”, or lexicographic, ordering

  45. A partial order on “induced” by partial orders on and on • Lexicographic ordering • Example:

  46. Well-Ordered Sets • A set S is well-ordered by the partial ordering if and only if every nonempty subset of S has a least element (minimum element). • Examples: • Non-Examples: , ,

  47. Hasse Diagrams • Begin with the digraph representation of the partial ordering • Omit the reflexive loops • Omit all edges which would be implied by transitivity • Orient all vertices and arcs so that the direction of each arc is up. • Remove the direction arrow from each arc

  48. Hasse Diagram Example I • Pairs in {1,2,3}{1,2,3} under lexicographic order

  49. Hasse Diagram Example I • Pairs in {1,2,3}{1,2,3} under lexicographic order

  50. Hasse Diagram Example I • Pairs in {1,2,3}{1,2,3} under lexicographic order

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