1. 931102fuzzy set theory chap07.ppt1 Fuzzy relations Fuzzy sets defined on universal sets which are Cartesian products.

2. 921028fuzzy set theory chap07.ppt2 Example (information retrieval) Documents D Key terms T Binary fuzzy relation R Membership degree R(d,t): the degree of relevance of document d and the key term t

3. 921028fuzzy set theory chap07.ppt3 Example (equality relation) y is approximately equal to x

4. 921028fuzzy set theory chap07.ppt4 Representations of fuzzy relation List of ordered pairs with their membership grades Matrices Mappings Directed graphs

5. 921028fuzzy set theory chap07.ppt5 Matrices Fuzzy relation R on X  Y X={x 1, x 2, …, x n }, Y={y 1, y 2, …, y m } r ij =R(x i, y j ) is the membership degree of pair (x i, y j )

6. 921028fuzzy set theory chap07.ppt6 Example (very far from) Example: X={Beijing, Chicago, London, Moscow, New York, Paris, Sydney, Tokyo} Very far from X

7. 921028fuzzy set theory chap07.ppt7 Mappings The visual representations On finite Cartesian products Document D = {d 1, d 2, …, d 5 } Key terms T = {t 1, t 2, t 3, t 4 }

8. 921028fuzzy set theory chap07.ppt8 Directed graphs

9. 921028fuzzy set theory chap07.ppt9 Inverse operation Given fuzzy binary relation R  X  Y –Inverse R -1  Y  X –R -1 (y,x) = R(x,y) –(R -1 ) -1 = R The transpose of R

10. 921028fuzzy set theory chap07.ppt10 The composition of fuzzy relations P and Q Given fuzzy relations P  X  Y, Q  Y  Z –Composition on P and Q = P  Q = R  X  Z –The membership degree of a chain is determined by the degree of the weaker of the two links, and. –R(x, z) = (P  Q )(x, z) = max y  Y min [ P(x, y ), Q(y, z )]

11. 921028fuzzy set theory chap07.ppt11 Example (composition of fuzzy relations) X = {a,b,c} Y = {1,2,3,4} Z = {A,B,C}

12. 921028fuzzy set theory chap07.ppt12 Example (matrix composition) P  X  Y Q  Y  Z R  X  Z

13. 921028fuzzy set theory chap07.ppt13 Fuzzy equivalence relations and compatibility relations Equivalence: Any fuzzy relation R that satisfies reflexive, symmetric and transitive properties. –R is Reflexive  R(x, x) = 1 for all x  X. –R is symmetric  R(x, y) = R(y, x) for all x, y  X. –R is transitive  R(x, z)  max y  Y min [ R(x, y ), R(y, z )] all x, z  X. Compatibility: satisfy reflexive and symmetric

14. 921028fuzzy set theory chap07.ppt14 Example (fuzzy equivalence)

15. 921028fuzzy set theory chap07.ppt15 Example(fuzzy compatibility) R(1,4)<max {min[R(1,2),R(2,4)], min[R(1,5),R(5,4)]}

16. 921028fuzzy set theory chap07.ppt16 Transitive closure The transitive closure of R is the smallest fuzzy relation that is transitive and contains R. interactive algorithm to find transitive closure R T of R 1.Compute R’ = R  (R  R) 2.If R’  R, rename R’ as R and go to step 1. Otherwise, R’= R T.

17. 921028fuzzy set theory chap07.ppt17 Evaluate the algorithm equal First iteration second iteration

18. 921028fuzzy set theory chap07.ppt18 Example (transitive closure) Transitive closure of R

19. 921028fuzzy set theory chap07.ppt19 Fuzzy Partial ordering Fuzzy relations that satisfy reflexive, transitive and anti-symmetric, denoted by x  y –R on X is anti-symmetric  R(x,y)>0 and R(y, x)>0 imply that x=y for any x,y  X.

20. 921028fuzzy set theory chap07.ppt20 Example (fuzzy partial ordering)

21. 921028fuzzy set theory chap07.ppt21 Projections example S ={s 1, s 2, s 3 } D ={d 1, d 2, d 3, d 4, d 5 } Projection on S Projection on D symptoms diseases

22. 921028fuzzy set theory chap07.ppt22 Projections Given an n-dimensional fuzzy relation R on X =X 1  X 2  …  X n and any subset P(any chosen dimensions) of X, the projection R P of R on P for each p  P is Where is the remaining dimensions

23. 921028fuzzy set theory chap07.ppt23 Cylindric extension Given an n-dimensional fuzzy relation R on X =X 1  X 2  …  X n, any (n+k)-dimensional relation whose projection into the n dimensions of R yields R is called an extension of R. An extension of R with respect to Y =Y 1  Y 2  …  Y h is call the cylindric extension EY R of R into Y EY R(x,y) = R(x) For all x  X and y  Y.

24. 921028fuzzy set theory chap07.ppt24 Example (Cylindric extension)

25. 921028fuzzy set theory chap07.ppt25 Example of cylindric closure cylindric closure : the intersection of the cylindric extensions of R is the original relation R.