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File: Matrix Pdf 174200 | 6 2019 12 29!10 48 43 Pm
3 3 fuzzy relation 71 it s corresponding fuzzy matrix is as follows a b c d a a 0 0 0 0 0 8 0 0 b 1 0 ...

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            §3.3 Fuzzy Relation                                                       71 
              It’s corresponding fuzzy matrix is as follows. 
                                      a b c d 
                          A
                           a 0.0 0.0 0.8 0.0
                           b 1.0 0.0 0.0 0.0
                           c 0.0 0.9 0.0 1.0
                           d 0.0 0.0 0.0 0.0
              Fuzzy relation is mainly useful when expressing knowledge. Generally, 
            the knowledge is composed of rules and facts. A rule can contain the 
            concept of possibility of event b after event a has occurred. For instance, 
            let us assume that set A is a set of events and R is a rule. Then by the rule 
            R, the possibility for the occurrence of event c after event a occurred is 0.8 
            in the previous fuzzy relation. 
              When crisp relation R represents the relation from crisp sets A to B, its 
            domain and range can be defined as, 
                            dom(R)   {x | x  A, y  A, P (x, y)   1} 
                                                       R
                            ran(R)   {y | x  A, y  A, P (x, y)   1} 
                                                       R
            Definition (Domain and range of fuzzy relation) When fuzzy relation R
            is defined in crisp sets A and B, the domain and range of this relation are 
            defined as :   
                                    P     (x)  max P (x, y)
                                     dom(R)         R
                                               yB
                                    P     (y)  max P (x, y)
                                      ran(R)        R
                                               xA
            Set A becomes the support of dom(R) and dom(R) Ž A. Set B is the support 
            of ran(R) and   
                                        ran(R) Ž B.  Ƒ
            3.3.3 Fuzzy Matrix  
            Given a certain vector, if an element of this vector has its value between 0 
            and 1, we call this vector a fuzzy vector. Fuzzy matrix is a gathering of 
             72                                           3. Fuzzy Relation and Composition 
             such vectors. Given a fuzzy matrix A   (a ) and B   (b ), we can perform 
                                                         ij            ij
             operations on these fuzzy matrices. 
                (1)  Sum 
                                         A + B   Max [a , b ]
                                                         ij  ij
                (2)  Max product 
                                  AxB AB  Max [ Min (a ,b ) ] 
                                                  k           ik  kj
                (3)  Scalar product 
                                         OA where 0 dOd 1 
             Example 3.4 The followings are examples of sum and max product on 
             fuzzy sets A and B.
                                 a b c                                    a b c 
              A =           a    0.2 0.5     0.0       B =          a     1.0   0.1 0.0 
                            b    0.4 1.0     0.1                    b     0.0   0.0 0.5 
                            c    0.0 1.0     0.0                    c     0.0   1.0 0.1 
                                 a b c                                    a b c 
              A + B =       a    1.0 0.5     0.0       AxB          a     0.2   0.1 0.5 
                            b    0.4 1.0     0.5                    b     0.4   0.1 0.5 
                            c    0.0 1.0     0.1                    c     0.0   0.0 0.5 
                Here let's have a closer look at the product A x B of A and B. For 
             instance, in the first row and second column of the matrix C   A x B, the 
             value 0.1 (C      0.1) is calculated by applying the Max-Min operation to 
                          12
             the values of the first row (0.2, 0.5 and 0.0) of A, and those of the second 
             column (0.1 , 0.0 and 1.0) of B.
                 0.2 0.5 0.0 
                 0.1 0.0 1.0 
                Min  
                 0.1 0.0 0.0 Ÿ 0.1 
                    Max
                In the same manner C13   0.5 is obtained by applying the same 
             procedure of calculation to the first row (0.2, 0.5, 0.0) of A and the third 
             column of B (0.0, 0.5, 0.1). 
                §3.3 Fuzzy Relation                                                       73 
                     0.2 0.5 0.0 
                     0.0 0.5 0.1 
                    Min  
                     0.0 0.5 0.0 Ÿ 0.5     Ƒ
                                                                   Max            
                    And for all i and j, if a d b  holds, matrix B is bigger than A.
                                                 ij     ij
                                                  aij d bij   œ           AdB
                    Also when A d B for arbitrary fuzzy matrices S and T, the following 
                relation holds from the Max-Product operation.   
                                          AdB œ                   SAdSB,ATdBT
                Definition (Fuzzy relation matrix) If a fuzzy relation R is given in the 
                form of fuzzy matrix, its elements represent the membership values of this 
                relation. That is, if the matrix is denoted by M , and membership values by 
                                                                             R
                P  (i, j), then    M   (P  (i, j))           Ƒ
                  R                    R       R
                3.3.4 Operation of Fuzzy Relation   
                We know now a relation is one kind of sets. Therefore we can apply 
                operations of fuzzy set to the relation. We assume R Ž A u B and 
                S Ž A u B.
                (1)    Union relation 
                      Union of two relations R and S is defined as follows : 
                             (x, y)  A u B
                                        P       (x, y)   Max [P  (x, y), P  (x, y)] 
                                          R‰S                       R           S
                                                         P  (x, y) › P  (x, y)
                                                            R             S
                       We generally use the sign › for Max operation. For n relations, we 
                    extend it to the following. 
                                                                 
                                             PR‰R ‰R ‰‰R x, y             ›PRi x,y
                                                1   2   3      n              R
                                                                               i
                       If expressing the fuzzy relation by fuzzy matrices, i.e. M  and M ,
                                                                                                      R           S
                    matrix M           concerning the union is obtained from the sum of two 
                                 R ‰ S
                    matrices M  + M .
                                   R      S
                                                       M        M + M
                                                         R‰S         R      S
                (2)    Intersection relation 
                    The intersection relation R ˆ S of set A and B is defined by the 
                    following membership function. 
              74                                           3. Fuzzy Relation and Composition 
                                     P      (x) =Min [P  (x, y), P  (x, y)] 
                                      Rˆ S               R          S
                                              = P  (x, y) šP (x, y) 
                                                   R            S
                    The symbol š is for the Min operation. In the same manner, the 
                 intersection relation for n relations is defined by 
                                       PR ˆR ˆR ˆˆR x, y             PRix, y
                                         1   2   3      n            š
                                                                     Ri
              (3)   Complement relation 
                  Complement relation  R  for fuzzy relation R shall be defined by the 
                  following membership function. 
                                   (x, y)  A u B P  (x, y)   1 - P  (x, y)
                                                        R               R
              Example 3.5 Two fuzzy relation matrices M  and M  are given. 
                                                                 R        S
                  M a  b  c                                     M a  b  c 
                    R                                              S
                  1 0.3 0.2 1.0                                 1 0.3 0.0 0.1 
                  2 0.8 1.0 1.0                                 2 0.1 0.8 1.0 
                  3 0.0 1.0 0.0                                 3 0.6 0.9 0.3 
              Fuzzy relation matrices M           and M        corresponding R ‰ S and R ˆ S 
                                             R‰S          Rˆ S
              yield the followings. 
                MR‰S     a b c  MRˆS a b c 
                1  0.3 0.2 1.0                                 1  0.3 0.0 0.1 
                2  0.8 1.0 1.0                                 2  0.1 0.8 1.0 
                3  0.6 1.0 0.3                                 3  0.0 0.9 0.0 
              Also complement relation of fuzzy relation R shall be 
                                         MR a  b  c 
                                         1  0.7 0.8 0.0 
                                         2  0.2 0.0 0.0 
                                         3  1.0 0.0 1.0 
              (4)   Inverse relation 
                 When a fuzzy relation R Ž A u B is given, the inverse relation of R-1 is 
                 defined by the following membership function.     
                  For all (x, y) Ž A u B,           P -1 (y, x)   P  (x, y)
                                                            R             R
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...Fuzzy relation it s corresponding matrix is as follows a b c d mainly useful when expressing knowledge generally the composed of rules and facts rule can contain concept possibility event after has occurred for instance let us assume that set events r then by occurrence in previous crisp represents from sets to its domain range be defined dom x y p ran definition this are max yb xa becomes support given certain vector if an element value between we call gathering composition such vectors perform ij operations on these matrices sum product axb ab k ik kj scalar oa where dod example followings examples here have closer look at first row second column calculated applying min operation values those same manner obtained procedure calculation third all i j holds bigger than aij bij adb also arbitrary t following sadsb atdbt form elements represent membership denoted m know now one kind therefore apply u union two relations rs use sign n extend prr pri e concerning intersection function symbo...

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