Section 9.1

1. Section 9.1 Relations and their Properties

2. Binary Relation from A to B Let A and B be sets. A binaryrelation from A to B is a subset of A x B.

3. Example Let A = { a, b, c } and B = { x, y } then A x B = {(a,x), (a,y), (b,x), (b,y), (c,x), (c,y)} Any subset of A x B is a relation from A to B. Therefore let, for example: R1 = {(a,x), (c,y)} R2 = {(a,y), (b,x)} ¿How many relations from A to B are there?

4. Binary Relation from A to B. A binary relation from A to B is a set R of ordered pairs where the first element of each ordered pair comes from A and the second element comes from B. aRb denotes that (a,b) is an element of R and aRb indicates that (a,b) is not an element of R. When (a,b) belongs to R then a is related tob by R.

5. Example Let: R1 = { (a,x), (c,y) } R2 = { (a,y), (b,x) } Notation a R1x b R1 x b R2 x

6. Depicting Relations • Directed Graphs • Tables • Example: • Suppose A = {0, 1, 2} and B = {a, b}. • Let R be the relation on A x B defined as: • {(0, a), (0, b), (1, a), (2, b)} • Diagrams on next slide

8. Functions as Relations A function is a relation from A to B such that every element in A is the first element of exactly one ordered pair of R.

9. Relations on a Set A relation on the set A is a relation from A to A. A relation on the set A is a subset of AxA

10. How Many Relations? If |A| = n then |AxA| = n2 . Since a relation is a subset of AxA then the number of possible relations is

11. Reflexive Relations Consider the following relations on the set {1,2,3,4} {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)} {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} {(2,4),(4,2)} {(1,2),(2,3),(3,4)} {(1,1),(2,2),(3,3),(4,4)} {(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}

12. Symmetric Relations Consider the following relations on the set {1,2,3,4} {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)} {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} {(2,4),(4,2)} {(1,2),(2,3),(3,4)} {(1,1),(2,2),(3,3),(4,4)} {(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}

13. Antisymmetric Relations Consider the following relations on the set {1,2,3,4} {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)} {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} {(2,4),(4,2)} {(1,2),(2,3),(3,4)} {(1,1),(2,2),(3,3),(4,4)} {(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}

14. Examples • The relation ≤ on the real numbers is antisymmetric • The relation ⊆ on sets is antisymmetric • The equality relation, =, on the real numbers is bothsymnmetric and antisymmetric • (antisymmetric ≠ asymmetric!) • Anther way to define antisymmetry: • aRb ∧a≠b →¬(bRa)

15. Transitive Relations Consider the following relations on the set {1,2,3,4} {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)} {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)} {(2,4),(4,2)} {(1,2),(2,3),(3,4)} {(1,1),(2,2),(3,3),(4,4)} {(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)}

16. Composition of Relations

17. Rn

18. Transitivity

19. Section 9.2 n-ary Relations

20. Definition Let A1, A2, … , An be sets. An n-ary relation on these sets is a subset of A1 x A2 x … x An. The sets A1, A2, … , An are called the domains of the relation, and n is called its degree.

21. Databases and Relations

22. Operations • Selection • Chooses tuples (rows) that satisfy a condition • Projection • Chooses slots (columns) by position number or name • Join • Combines tuples from two tables with a common key

23. Relational Algebra Demo • reldb.py • 4 Tables: • parts • suppliers • projects • spj (a table that connects the other 3 tables)

24. Section 9.3 Representing Relations

25. Using Matrices Let A = {1,2,3} and B={1,2}. Let the relation from A to B containing (a,b) if a>b. R would then equal {(2,1), (3,1), (3,2)} 1 2 1 2 3

26. Reflexive Relations

27. Symmetric SYMMETRIC NOTSYMMETRIC

28. Antisymmetric

29. Boolean Matrices Join of Matrices Union of Relations Meet of Matrices Intersection of Relations Boolean Product of Matrices Composite Relations

30. a b V = {a,b,c,d} (a,b), (a,d), (c,b), (c,a), d c (b,b), (d,b), (b,d)} Digraph Example E = {

31. Spotting Properties via Graphs • How can you spot the following properties from a graph? • Reflexive • Symmetric • Antisymmetric • Transitive

33. Section 9.4 Closures of Relations

34. Reflexive Closure • The reflexive closure of a relation, R, is the minimal relation containing R that is reflexive • If R is already reflexive, it is its own reflexive closure • Otherwise, add pairs (a,a) for all elements a, that are not already related to themselves. • For example, if R = {(1,1), (1,2), (2,1), (3,2)}, the we add the following pairs: (2,2), (3,3).

35. Symmetric Closure • The symmetric closure of a relation, R, is the minimal relation containing R that is symmetric • If R is already symmetric, it is its own symmetric closure • Otherwise, add pairs (b,a) for all pairs (a,b) in R that are not already there. • For example, if R = {(1,1), (1,2), (2,1), (3,2)}, the we add the following pair: (2,3).

36. Can You Get There From Here? • Often we want to know if there is a “path” from a to b (meaning a chain of pairs (a,x1), (x1,x2), … (xn,b)). • We say that b is “reachable in R” from a. • Forming this relation is harder than it appears

37. Transitive Closure • The transitive closure of a relation, R, is the minimal relation containing R that is transitive. • Theorem 1: There is a path from a to b in R of length n, if and only if (a,b) ⋲ Rn. (The composition of R with itself n-1 times). • R defines paths of length 1 • R2 adds paths of length 2, and so on…

38. Computing the Transitive Closure • Rn computes paths of length n. This means that n+1 objects have been visited, forcing a repeat (cycle). So everything has been traversed! • So all possible connected endpoints are computed by

39. Matrix Representation • M* = M v M2 v M3 v … v Mn. • See Example 7 in section 9.4, page 602-603. • A more efficient algorithm (Warshall’s algorithm) is on page 606 (Algorithm 2 in section 9.4).

40. Section 9.5 Equivalence Relations

41. Equivalence Relations A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.

42. Examples: Let R be the relation on the set of strings such that a R b if and only if l(a) = l (b) where l (x) is the length of x. Let R be the relation on the set of real numbers such that a R b if and only if a-b is an integer.

43. Is this an equivalence relation? Which elements are “equivalent” (related to each other)?

44. 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 will delete the subscript R and write [a] for this equivalence class.

45. [0] = {…-14, -7, 0, 7, 14, 21,…} [1] = {…-13, -6, 1, 8, 15, 22,…} [2] = {…-12, -5, 2, 9, 16, 23,…} [3] = {…-11, -4, 3, 10, 17, 24,…} [4] = {…-10, -3, 4, 11, 18, 25,…} [5] = {…-9, -2, 5, 12, 19, 26,…} [6] = {…-8, -1, 6, 13, 20, 27,…}

47. Partition A partition of a set S is a collection of disjoint, nonempty subsets of S that have S as their union.

48. Equivalence Relations and Partitions Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition of the set S, there is an equivalence relation R that has the sets as its equivalence classes.

49. Section 9.6 Partial Orderings

50. Partially Ordered Set (POSET) 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)