Classical relations and fuzzy relations

1. Classical Relations And Fuzzy Relations Baran Kaynak 1

2. Relations This chapter introduce the notion of relation. The notion of relation is the basic idea behind numerous operations on sets s uch as Cartesian products, composition of relations , difference of relations and intersections of relations and equivalence properties In all engineering , science and mathematically based fields, relations is very important 2

3. Relations Similarities can be described with relations. In this sense, relations is a very important notion to many different technologies like graph theory , data manipulation . Graph theory 3

4. Data manipulations 4

5. In classical relations ( crisp relations ), Relationships between elements of the sets are only in two degrees ; “ completely related ” and “not related ”. Fuzzy relations take on an infinitive number of degrees of relationships between the extremes of “ completely related ” and “ not related ” 5

6. Crisp system Crisp, exact - B ased on models (i.e. differential equations) - Requires complete set of data - Typically linear Fuzzy system - Fuzzy, qualitative, vague - Uses knowledge (i.e. rules) - Requires fuzzy data - Nonlinear method 6

7. Crisp system -Complex systems hard to model -incomplete information leads to inaccuracy -numerical Fuzzy logic system -No traditional modeling, inferences based on knowledge - can handle incomplete information to some degree -linguistic 7

8. Example 3.1. The elements in two sets A and B are given as A ={0, 1} and B ={ a,b , c}. Various Cartesian products of these two sets can be written as shown: A × B ={(0,a),(0,b),(0,c),(1,a),(1,b),(1,c)} B × A ={(a, 0), (a, 1), (b, 0), (b, 1), (c, 0), (c, 1)} A × A = A 2 ={(0, 0), (0, 1), (1, 0), (1, 1)} B × B = B 2 ={(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)} Cartesian Product 8

9. Crisp Relations Cartesian product is denoted in form A1 x A2 x…..x Ar The most common case is for r=2 and represent with A1 x A2 The Cartesian product of two universes X and Y is determined as X × Y = {(x, y) | x ∈ X,y ∈ Y} This form shows that there is a matching between X and Y , this is a unconstrained matching . 9

10. Crisp Relations E very element in universe X is related completely to every element in universe Y This relationship’s strenght is measured by the characteristics function χ χX×Y(x, y) = 1, (x,y) ∈ X × Y 0, (x,y) ∉ X × Y Complete relationship is showed with 1 and no relationship is showed with 10

11. When the universes, or sets, are finite the relation can be conveniently represented by a matrix, called a relation matrix. X ={1, 2, 3} and Y ={a, b, c} Sagittal diagram of an unconstrained relation 11

12. Special cases of the constrained Cartesian product for sets where r=2 are called identity relation denoted I A I A ={(0, 0), (1, 1), (2, 2)} Special cases of the unconstrained Cartesian product for sets where r=2 are called universal relation denoted U A U A ={(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)} 12

13. Cardinality Of Crips Relations The Cardinality of the relation r between X and Y is n X x Y = nx * ny Power set (P(X x Y)), n P(X×Y) = 2 ( nXnY ) 13

14. Operations On Crips Relations Define R and S as two separate relations on the Cartesian universe X × Y Union : R ∪ S → χ R∪S(x, y) : χ R∪S(x, y) = max[ χ R(x, y), χ S(x, y)] Intersection : R ∩ S → χ R∩S(x, y) : χ R∩S(x, y) = min[ χ R(x, y), χ S(x, y)] Complement : R → χ R(x, y) : χ R(x, y) = 1 − χ R(x, y) Containment : R ⊂ S → χ R(x, y) : χ R(x, y) ≤ χ S(x, y) 14

15. Properties Of Crips Relations Commutativity A ssociativity D istributivity Involution I dempotency 15 2 × (1 + 3) = (2 × 1) + (2 × 3).

16. Composition 16

17. Crisp Binary Relation 17

18. Composition For these two relations lets make a composition named T R = {(x1, y1), (x1, y3), (x2, y4)} S = {(y1, z2), (y3, z2 )} 18

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20. A chain is only as strong as its weakest link 20

21. Example µ T (x1 , z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 U sing the max–min composition operation , r elation matrices for R and S would be expressed as 21

22. Example U sing the max–min composition operation , r elation matrices for R and S would be expressed as µ T (x1 , z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 µ T (x1 , z 2 ) = max[min ( 1 , 1 ), min ( , ), min ( 1 , 1 ), min ( , ) ] = 1 22

23. Fuzzy Relations A fuzzy relation R is a mapping from the Cartesian space X x Y to the interval [0,1], where the strength of the mapping is expressed by the membership function of the relation μR ( x,y ) μ R : A × B → [0, 1] R = {(( x , y ), μ R ( x , y ))| μ R ( x , y ) ≥ 0 , x ∈ A , y ∈ B } 23

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25. Crisp relation vs. Fuzzy relation 25 Crisp relation Fuzzy relation

26. Cardinality of Fuzzy Relations Since the cardinality of fuzzy sets on any universe is infinity, the cardinality of a fuzzy relation between two or more universes is also infinity. 26

27. Operations on Fuzzy Relations Let R and S be fuzzy relations on the Cartesian space X × Y. Then the following operations apply for the membership values for various set operations: 27 Union : µR∪S (x, y) = max(µR (x, y),µS(x, y)) Intersection : µR∩S (x, y) = min(µR (x, y),µS (x, y)) Complement : µR(x, y) = 1 − µR(x, y) Containment : R⊂ S ⇒ µR (x, y) ≤ µS (x, y)

28. Fuzzy Cartesian Product and Composition A fuzzy relation R is a mapping from the Cartesian space X x Y to the interval [0,1], where the strength of the mapping is expressed by the membership function of the relation μR ( x,y ) μ R : A × B → [0, 1] R = {(( x , y ), μ R ( x , y ))| μ R ( x , y ) ≥ 0 , x ∈ A , y ∈ B } 28

29. Max-min Composition Two fuzzy relations R and S are defined on sets A, B and C . That is, R ⊆ A × B, S ⊆ B × C. The composition S•R = SR of two relations R and S is expressed by the relation from A to C : For ( x , y ) ∈ A × B , ( y , z ) ∈ B × C , µ S•R (x, z) = max [ min ( µ R (x, y), µ S (y, z ))] = ∨ [μ R ( x , y ) ∧ μ S ( y , z )] M S • R = M R • M S ( matrix notation ) 29

30. Max-min Composition 30

31. Max-product Composition Two fuzzy relations R and S are defined on sets A , B and C . That is, R ⊆ A × B, S ⊆ B × C. The composition S • R = SR of two relations R and S is expressed by the relation from A to C: For ( x , y ) ∈ A × B , ( y , z ) ∈ B × C , μ S • R ( x , z ) = max y [ μ R ( x , y ) • μ S ( y , z )] = ∨ y [μ R ( x , y ) • μ S ( y , z ) M S • R = M R • M S ( matrix notation ) 31

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33. Example Suppose we have two fuzzy sets, A defined on a universe of three discrete temperatures , X = { x 1 , x 2 , x 3}, and B defined on a universe of two discrete pressures, Y = { y 1 , y 2}, and we want to find the fuzzy Cartesian product between them. Fuzzy set A could represent the ‘‘ambient’’ temperature and f uzzy set B the ‘‘near optimum’’ pressure for a certain heat exchanger, and the Cartesian product might represent the conditions ( temperature–pressure pairs ) of the exchanger that are associated with ‘‘efficient’’ operations. 33

34. Fuzzy Cartesian product , using µ S•R (x, z) = max [ min (µ R (x, y), µ S (y, z ))] results in a fuzzy relation R (of size 3 × 2) representing ‘‘efficient’’ conditions , 34

35. Example X = { x 1 , x 2} , Y = { y 1 , y 2} , and Z = { z 1 , z 2 , z 3} Consider the following fuzzy relations: 35 Then the resulting relation , T , which relates elements of universe X to elements of universe Z , μ T (x 1 , z 1 ) = max [ min ( . 7 , . 9 ), min ( . 5 , . 1 ) ] = 0 . 7

36. and by max – product composition , 36 μ T (x2 , z2) = max[(0.8 . 0.6 ), (0.4 . 0.7)] = 0.48

37. Example A simple fuzzy system is given, which models the brake behaviour of a car driver depending on the car speed. The inference machine should determine the brake force for a given car speed. The speed is specified by the two linguistic terms " low " and " medium " , and the brake force by " moderate " and " strong " . The rule base includes the two rules (1) IF the car speed is low THEN the brake force is moderate (2) IF the car speed is medium THEN the brake force is strong 37

38. http:// virtual.cvut.cz/dynlabmodules/ihtml/dynlabmodules/syscontrol/node123.html Short Url : http ://goo.gl/T2z5u 38

39. Crisp Equivalence Relation A relation R on a universe X can also be thought of as a relation from X to X. The relation R is an equivalence relation and it has the following three properties: Refl exivity Symmetry Transitivity 39

40. Reflexivity (xi ,xi ) ∈ R or χ R(xi ,xi ) = 1 When a relation is reflexive every vertex in the graph originates a single loop, as shown in 40

41. Symmetry (xi, xj ) ∈ R → (xj, xi) ∈ R 41

42. Transitivity (xi , xj ) ∈ R and ( xj , xk ) ∈ R → (xi , xk ) ∈ R 42

43. Crisp Tolerance Relation A tolerance relation R (also called a proximity relation) on a universe X is a relation that exhibits only the properties of reflexivity and symmetry. A tolerance relation, R, can be reformed into an equivalence relation by at most (n − 1) compositions with itself , where n is the cardinal number of the set defining R, in this case X 43

44. Example Suppose in an airline transportation system we have a universe composed of five elements: the cities Omaha, Chicago, Rome, London, and Detroit. The airline is studying locations of potential hubs in various countries and must consider air mileage between cities and takeoff and landing policies in the various countries. 44

45. Example These cities can be enumerated as the elements of a set, i.e., X ={x1,x2,x3,x4,x5}={Omaha, Chicago, Rome , London, Detroit} S uppose we have a tolerance relation, R1, that expresses relationships among these cities: This relation is reflexive and symmetric. 45

46. Example The graph for this tolerance relation If (x1,x5) ∈ R1 can become an equivalence relation 46

47. Example : This matrix is equivalence relation because it has (x1,x5) 47 Five-vertex graph of equivalenc e relation ( reflexive , symmetric, transitive)

48. FUZZY TOLERANCE AND EQUIVALENCE RELATIONS Reflexivity μ R (xi , xi) = 1 Symmetry μ R (xi , xj ) = μ R ( xj , xi ) Transitivity μ R (xi , xj ) = λ 1 and μ R ( xj , xk ) = λ 2 μ R (xi , xk ) = λ where λ ≥ min [ λ 1 , λ 2 ]. 48

49. Example Suppose, in a biotechnology experiment, five potentially new strains of b acteria have been detected in the area around an anaerobic corrosion pit on a new aluminum - lithium alloy used in the fuel tanks of a new experimental aircraft. In order to propose methods to eliminate the biocorrosion caused by these bacteria, the five strains must first be categorized. One way to categorize them is to compare them to one another. In a pairwise comparison, the following " similarity " relation,R1 , is developed. For example, the first strain (column 1) has a strength of similarity to the second strain of 0.8, to the third strain a strength of 0 (i.e., no relation ), to the fourth strain a strength of 0.1, and so on. Because the relation is for pairwise similarity it will be reflexive and symmetric. Hence, 49

50. 50 is reflexive and symmetric. However, it is not transitive μ R (x1 , x2) = 0.8, μ R (x2 , x5) = 0.9 ≥ 0.8 but μ R (x1 , x5) = 0.2 ≤ min (0.8 , 0.9)

51. 51 One composition results in the following relation: where transitivity still does not result; for example, μ R 2 (x 1 , x 2 ) = 0 . 8 ≥ 0 . 5 and μ R 2 (x 2 , x 4 ) = 0 . 5 but μ R 2 (x 1 , x 4 ) = 0 . 2 ≤ min ( . 8 , . 5 )

52. 52 Finally, after one or two more compositions, transitivity results: