# Fuzzy relation Fuzzy relation

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Fuzzy relation If U is the Cartesian product of n universes of discourse U1,…, Un, that is, U=U1 x…x Un, then an n-array fuzzy relation, R, in U is a fuzzy subset of U and is defined is R  R (u1, , un ) /(u1, , un )  R (u1 , , u n ) ,where function R is the membership Example : Consider U1 U ( ,) Define the relation “close to” by e   u1  u2 /(u1 , u2 )  u1 xu where  is a scalar factor Similarly, for u1 than the 2relation U1, U 1,2,3“,4 is much greater u2 by the relation matrix can be defined R 4 0.3 0.8 0 0.3 0.8 0 0.3 0 0 ” Or, equivalently, 0.0 0.3 0.8 1.0 0.0 0.0 0.3 0.8 R= 0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.0 Or we can use a table (called fuzzy matrix) to define the relation, instead of a algebraic function Ex: Let V  x1 , x2 , x3  , Y  y1 , y2 , y3 , y4  R= “x is considerably larger than y” y1 y2 x1 0.8 x2 y3 0.1 0.7 0.8 X3 0.9 y4 0.7 0.8 Z= “y is very close to x”  Fuzzy matrix y1 y2 y3 y4 x1 0.4 0.9 0.6 x2 0.9 0.4 0.5 0.7 X3 0.3 0.8 0.5 Operations of Relations Union  R Z ( x, y ) Max  R ( x, y ),  Z ( x, y ) where ( x, y )  V xY Intersection  R Z ( x, y ) Min  R ( x, y ),  Z ( x, y ) where ( x, y )  X xY Complement Contained  B ( x, y ) 1   B ( x, y ) R  Z  R ( x , y )  Z ( x , y ) y1 RZ RZ y2 y3 y4 x1 0.8 x2 0.9 0.8 0.5 0.7 X3 0.9 0.8 0.8 y1 y3 y2 x1 0.4 x2 X3 0.3 0.9 0.7 y4 0.1 0.6 0.4 0 0.7 0.5 Converse R  R  ( y , x )  R ( x , y ) Identity relation  x y  I ( x, y )  if 0 x y  Z ( x, y ) 0 x  X , y  Y Null relation Z  Z ( x, y ) 1 x  X , y  Y Universal relation U Superstar Composition ROS =  x, z , sup  R ( x, y ) *  s ( y , z )  | x  X , y  Y , z  Z  Composition of Fuzzy Relations *:can be any of TNorm Max - Min Composition Let R1 ( x, y ) ( x, y )  X xY and R2 ( y , z ) ( x, y )  Y x Z be two fuzzy relation The max - composition R1 maximin R2 is the fuzzy relation R10 R2  ( x, z ), max min  R ( x, y ),  R ( y , z )   | x  X , y  Y   y max   max - product R10 R2  ( x, z ), max   R1 ( x, y ),  R ( y , z )   | x  X , y  Y   y   max - average Composition    R10 R2  ( x, z ), max   R1 ( x, y )   R ( y, z )   | x  X , y  Y       Linear alegbra R1xR2 ( x, z ) sup  R1   n      Property of the Max-Min Composition Associatively Reflectivity ( R3  R2 )  R1 R3  ( R2  R1 ) Let R be a fuzzy relation in X x X R is called reflexive if  R ( x, x) 1 x  X E reflective  R ( x, x)  x  X    Symmetry R( x, y ) R ( y, z ) R  ( x, y ) Example 3.21 Given a fuzzy set A in X and a fuzzy relation R in X X Y as follows, x1 x2 x3 A 0.2 x1  0.8 x  x  0.2 0.8 1 , x1 R= x2 y1 0.7 0.5 0.2 y2 0.9 0.6 y3 0.4 0.6 0.3 x3 y1 x1 0.7 x1 x2 x3 B = A 。 R=(0.2 0.8 1) 。 x2 x0.5 0.2 0.5 y1  0.8 y2  0.6 y3 y2 y3 y y y3 0.4 0.9 0.6 = (0.5 0.8 0.6) 0.6 0.3 Next, we shall study the following more difficult problems: Fuzzy Input A Fuzzy System R Fuzzy Output B Figure 3.5 Fuzzy system with fuzzy input and fuzzy output Next, we shall study the following more difficult problems: P1: Given A and B, determine R such that A 。 R =B P2: Given R and B, determine A such that A 。 R =B Theorem 3.1 Problem P1 has solution(s) if and only if the height of the fuzzy set A is greater than or equal to the height of the fuzzy set B, that is, max  A ( x )  B ( y ) for all y Y xX In order to solve problem P1 (and problem P2 later), we need to introduce the -operation For any a, b﹝0,1﹝, the -operator is defined as if a b (3.50) a b   b if a b   For fuzzy sets A and B in X and Y, respectively, the -composition of A and B forms a fuzzy relation A srB in X x Y which is defined by   AsuBur  x, y   A  x   B  y   B  y  if  A ( x )  B  y  if  A ( x )  B  y    Furthermore, the -composition of a fuzzy relation R and a fuzzy set B is denoted by R srB and is defined by    RsuBur  x     R  x, y   B  y   yY With the above -composition, the following properties will be useful for determining the solutions of problem P1 Theorem 3.2 Let R be a fuzzy relation on X x Y For any fuzzy sets A and B in X and Y, respectively, we have R  A sr  A oR  , (3.53) A o A srB   B (3.54) Theorem 3.3 If the solution of problem P1 exists, then the large R (in the sense of set-theoretic inclusion th at satisfies the fuzzy relation equation A 。 R =B i s Rˆ Rˆ  A srB (3.55) Example 3.22 given A 0.2 x1  0.8 x2  x3 and B 0.5 y1  0.8 y2 ,then we have Rˆ  A srB x1 0.2 x2 0.8 x3 x1 y y2 y3 = x2 (0.5 0.8 0.6) x3 y1 y2 y3 1 0.5 0.6 0.5 0.8 0.6 Theorem 3.4 Problem P2 has no solution if the following inequality holds: max  R  x, y    B  y  for some y  Y xX (3.56) Example 3.23 Consider the max-min composition of A 。 R: y1 y2 x1 0.8 0.5 x1 ,x2 ,x3 。 x2 0.7 = 0.5,0.1 =B x3 0.3 0.2 Since max xX  R  x, y2  max  0.5,0.7,0.2 0.7   B  y2  =0.1 , this set-relation equation has no solution Consider another fuzzy set-relation equation: y1 y2 x1 1 x ,x ,x A。R= 1 x2 = 0.5 , 0.7 =B x3 1 Although max xX  R  x, y1  1   B  y1  =0.5 and there is obviously no solution to the above set-relation equation max xX  R  x, y2  1   B  y2  =0.7 , Theorem 3.5 Let R be a fuzzy relation on X x Y For any fuzzy sets A and B in X and Y, respectively, we have  RsrB  oR  B, A  R sr  A oR  (3.57) (3.58) Theorem 3.6 If a solution to problem P2 exists, then the larges t fuzzy set A that satisfies A 。 R=B is : Aˆ Whose membership function is given by Eq (3.5 2) Aˆ R srB,  3.59  Example 3.24 From Example 3.21, we are given R and B Then, have y1 y2 y3 x1 Rˆ  A srB = x2 x3 0.7 0.4 0.5 0.9 0.6 0.2 0.6 0.3 y1 y2 y3 0.5 0.8 0.6  0.5  0.8    0.5    0.8     1.8       1         From the solution Aˆ that we obtained, we can confirm that Aˆ 。 R = B and A Aˆ   ... 0.5 0.8   0.4  Fuzzy Relation Equation Let A be a fuzzy set in X and R(X,Y) be a binary fuzzy r elation in X x Y The set -relation composition of A and R 。 A, results in a fuzzy set in Y Let... shall study the following more difficult problems: Fuzzy Input A Fuzzy System R Fuzzy Output B Figure 3.5 Fuzzy system with fuzzy input and fuzzy output Next, we shall study the following more... important types of binary fuzzy relations are distinguished on the basis of three different characteristic properties: reflexivity, symmetry, and transitivity A fuzzy relation R(X,X) is reflexive
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