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 suchas 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. Graphtheory 3

4. Data manipulations 4

5. Inclassicalrelations (crisprelations), Relationshipsbetweenelements of thesetsare only in twodegrees; “completelyrelated” and “not related”. Fuzzyrelationstake on an infinitivenumber of degrees of relationshipsbetweentheextremes of “ completelyrelated” and “ not related” 5

7. Crisp system -Complex systems hardto 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. CartesianProduct 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 bewritten 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 = A2={(0, 0), (0, 1), (1, 0), (1, 1)} B × B = B2={(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)} 8

9. CrispRelations Cartesianproduct is denoted in form A1 x A2 x…..x Ar Themostcommoncase is for r=2 andrepresentwith 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 showsthatthere is a matchingbetween X and Y , this is a unconstrainedmatching. 9

10. CrispRelations Every element in universe X is related completely toevery element in universe Y Thisrelationship’sstrenght is measuredbythecharacteristicsfunctionχ χX×Y(x, y) = 1, (x,y) ∈ X × Y 0, (x,y) ∉ X × Y Completerelationship is showedwith 1 and no relationship is showedwith 0 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. Specialcases of theconstrainedCartesianproductforsetswhere r=2 arecalledidentityrelationdenoted IA IA ={(0, 0), (1, 1), (2, 2)} Specialcases of theunconstrainedCartesianproductforsetswhere r=2 arecalleduniversalrelationdenoted UA UA ={(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)} 12

13. Cardinality Of CripsRelations TheCardinality of therelation r between X and Y is n X x Y = nx * ny Power set (P(X x Y)), nP(X×Y) = 2(nXnY) 13

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

15. Properties Of CripsRelations Commutativity Associativity Distributivity Involution Idempotency 15 2 × (1 + 3) = (2 × 1) + (2 × 3).

16. Composition 16

17. CrispBinaryRelation 17

18. Composition Forthesetworelationsletsmake a compositionnamed 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 itsweakestlink 20

21. Example Using themax–min composition operation,relationmatrices for Rand S would be expressed as µT(x1, z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 21

22. Example Using themax–min composition operation,relationmatrices for Rand S would be expressed as µT(x1, z1) = max[min(1, 0), min(0, 0), min(1, 0), min(0, 0)] = 0 µT(x1, z2) = max[min(1, 1), min(0, 0), min(1, 1), min(0, 0)] = 1 22

23. FuzzyRelations A fuzzy relation R is a mapping from the Cartesianspace X x Y to the interval [0,1], where thestrength of the mapping is expressed by themembership function of the relation μR(x,y) μR : A × B -> [0, 1] R = {((x, y), μR(x, y))| μR(x, y) ≥ 0 , x ∈ A, y ∈ B} 23

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25. Crisp relation vs. Fuzzyrelation 25 Crisprelation Fuzzyrelation

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

27. Operations on FuzzyRelations Let R and S be fuzzy relations on the Cartesian space X × Y. Then the following operationsapply 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 Cartesianspace X x Y to the interval [0,1], where thestrength of the mapping is expressed by themembership function of the relation μR(x,y) μR: A × B -> [0, 1] R = {((x, y), μR(x, y))| μR(x, y) ≥ 0 , x ∈ A, y ∈ B} 28

29. Max-minComposition Two fuzzy relations R and S are defined on sets A,B and C. That is, R ⊆ A × B, S ⊆ B × C. Thecomposition S•R = SR of two relations R and S isexpressed by the relation from A to C: For(x, y) ∈ A × B, (y, z) ∈ B × C, µS•R(x, z) = max [min(µR(x, y), µS(y, z))]= ∨ [μR(x, y) ∧ μS(y, z)] MS•R= MR•MS(matrixnotation) 29

30. Max-minComposition 30

31. Max-productComposition Two fuzzy relations R and S are defined on sets A,B and C. That is, R ⊆ A × B, S ⊆ B × C. Thecomposition S•R = SR of two relations R and S isexpressed by the relation from A to C: For(x, y) ∈ A × B, (y, z) ∈ B × C, μS•R(x, z) = maxy[μR(x, y) • μS(y, z)] = ∨y[μR(x, y) • μS(y, z) MS•R= MR• MS(matrixnotation) 31

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33. Example Suppose we have two fuzzy sets, Adefined on a universe of three discretetemperatures, X = {x1, x2, x3}, and Bdefined on a universe of two discrete pressures, Y ={y1, y2}, and we want to find the fuzzy Cartesian product between them. Fuzzy set Acouldrepresent the ‘‘ambient’’ temperature and fuzzy setBthe ‘‘near optimum’’ pressure for a certainheat exchanger, and the Cartesian productmight represent the conditions (temperature–pressurepairs) of the exchanger that are associated with ‘‘efficient’’ operations. 33

34. FuzzyCartesianproduct, usingµS•R(x, z) = max [min (µR (x, y), µS (y, z))]results in a fuzzyrelation R (of size 3 × 2) representing ‘‘efficient’’ conditions, 34

35. Example X = {x1, x2}, Y = {y1, y2}, and Z = {z1, z2, z3} Consider the following fuzzy relations: 35 Thentheresultingrelation, T, which relates elements of universe X to elements of universe Z, μT(x1, z1) = max[min(0.7, 0.9), min(0.5, 0.1)] = 0.7

36. andbymax–productcomposition, 36 μT(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. CrispEquivalenceRelation Arelation R on a universeXcan also be thought of as a relation fromXtoX. The relation R is an equivalence relation and it has the following three properties: Reflexivity Symmetry Transitivity 39

40. Reflexivity (xi ,xi ) ∈ R or χR(xi ,xi ) = 1 When a relation is reflexiveevery 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. CrispToleranceRelation 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 offive elements: the cities Omaha, Chicago, Rome, London, and Detroit. The airline is studyinglocations of potential hubs in various countries and must consider air mileage between citiesand 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} Suppose 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) ∈ R1can become an equivalence relation 46

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

48. FUZZY TOLERANCE AND EQUIVALENCE RELATIONS Reflexivity μR(xi, xi) = 1 Symmetry μR(xi, xj ) = μR(xj, xi) Transitivity μR(xi, xj ) =λ1 and μR(xj, xk) = λ2 μR(xi, xk) =λwhereλ ≥ min[λ1, λ2]. 48

49. Example Suppose, in a biotechnology experiment, five potentially new strains of bacteriahave been detected in the area around an anaerobic corrosion pit on a new aluminum-lithiumalloy used in the fuel tanks of a new experimental aircraft. In order to propose methods toeliminate 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, thefollowing " similarity" relation,R1, is developed. For example, the first strain (column 1) hasa strength of similarity to the second strain of 0.8, to the third strain a strength of 0 (i.e., norelation), to the fourth strain a strength of 0.1, and so on. Because the relation is for pairwisesimilarity it will be reflexive and symmetric. Hence, 49

50. 50 is reflexive and symmetric. However, it is not transitive μR(x1, x2) = 0.8, μR(x2, x5) = 0.9 ≥ 0.8 but μR(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, μR2(x1, x2) = 0.8 ≥ 0.5 and μR2(x2, x4) = 0.5 but μR2(x1, x4) = 0.2 ≤ min(0.8, 0.5)

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