Chapter 4 Fuzzy Graph and Relation. 4.1.1 Graph and Fuzzy Graph Graph G  (V, E) V : Set of vertices(node or element) E : Set of edges An edge is pair.

Chapter 4 Fuzzy Graph and Relation

4.1.1 Graph and Fuzzy Graph Graph G  (V, E) V : Set of vertices(node or element) E : Set of edges An edge is pair (x, y) of vertices in V. Fuzzy Graph V : set of vertices E : fuzzy set of edges between vertices

4.1.2 Fuzzy graph and fuzzy relation Fig 4.1 Fuzzy graph Example 4.1 (an example of fuzzy graph) MGMG b1b1 b2b2 a1a1 0.80.2 a2a2 0.30.0 a3a3 0.70.4

4.1.2 Fuzzy graph and fuzzy relation Fig 4.1 Fuzzy graph y  x ( y closes to x) Example 4.2   : nonnegative real numbers. x    and y    R  {(x, y) | x  y}, R      .

4.1.2 Fuzzy graph and fuzzy relation Fig 4.3 Fuzzy relation and fuzzy graph Example 4.3 The darkness of color stands for the strength of relation in (a) Relation (a, b) is stronger than that of relation (a, c). The corresponding fuzzy graph is shown in (b). the strength of relation is marked by the thickness of line. (a) Fuzzy relation R(b) Fuzzy graph

4.1.2 Fuzzy graph and fuzzy relation Fig 4.4 Fuzzy graph Example 4.4 b1b1 B2B2 b3b3 a1a1 0.51.00.0 a2a2 0.5 a3a3 1.0 0.0 mapping function  (A)  {a 1 }  {(b 1, 0.5), (b 2, 1.0)}  {a 2 }  {(b 3, 0.5)}  {a 3 }  {(b 1, 1.0), (b 2, 1.0)}  {a 1, a 2 }  {(b 1, 0.5), (b 2, 1.0), (b 3, 0.5)}

4.1.2 Fuzzy graph and fuzzy relation Fig 4.5 Fuzzy graph Example 4.5 A picture and fuzzy relation Fig 4.6 Fuzzy graph(by coordinates)

4.1.2 Fuzzy graph and fuzzy relation (a) Graph  R (x, y)  x 2 + y 2  1 Example 4.6 A graph and a fuzzy graph (b) Graph  R (x, y)  x 2 + y 2  1 1 1 1 y x 1 1 x y Fig 4.7 Fuzzy graph

4.1.3  -cut of Fuzzy Graph Example 4.7 Appling  -cut operation on fuzzy graph, for example A  {a, b, c}, R  A  A is defined as follows. abc a1.00.80.4 b0.00.40.0 c0.81.00.0 b a c

4.1.3  -cut of Fuzzy Graph abc a 1.0 b 0.01.00.0 c 1.0 0.0 abc a 1.0 0.0 b c 1.0 0.0 abc a 1.00.0 b c 1.00.0

4.1.3  -cut of Fuzzy Graph Fig 4.9 Graphical form of R Example 4.8  R (x, y) = x/2 + y  1 Fig 4.10 Graphical representation of R 0.5

4.1.3  -cut of Fuzzy Graph Fig 4.11 Set  A (x)= x Example 4.9  A (x) = x  R (x,y) = x+y  1, x  A, 0  y  1 Fig 4.12 Relation  A (x,y)= x+y  1, x  A

4.1.3  -cut of Fuzzy Graph Example 4.10 A={ x | x close to 2k , k = -1,0, 1,2, ….}  A (x) = Max[0, cosx].  A (x) x Fig 4.13 Set  A (x)=cosx  0 Fig 4.14  -cut set A 0.5

4.1.3  -cut of Fuzzy Graph Fig 4.15 Relation  R (x,y)=cosx. Fig 4.16  -cut relation R 0.5

4.1.4 Fuzzy Network Path with fuzzy edge V : a crisp set of nodes, R : a relation defined on the set V path C i  (x i1, x i2,..., x ir ), x ik  V, k  1, 2,..., r where  (x ik, x ik  1 ),  R (x ik, x ik  1 )  0, k  1, 2,..., r-1 fuzzy value l for path C i : the minimum possibility of connecting from x i1 to x ir. l (x i1, x i2,..., x ir )   R (x i1, x i2 )   R (x i2, x i3 ) ...   R (x ir-1, x ir ) possible set of paths C(x i, x j )  {c(x i, x j ) | c(x i, x j )  (x i1  x i, x i2,..., x ir  x j )} value of maximum intensity path l* (x i, x j )   l (x i1  x i, x i2,..., x ir  x j ) C(xi, xj)

4.1.4 Fuzzy Network (b) Fuzzy network (node,edge) Path with fuzzy node and fuzzy edge V : a fuzzy set of nodes, R : a fuzzy set of edge C i  (x i1, x i2,..., x ir ), x ik  V, k  1, 2,..., r where,  ( x ik, x ik  1 ),  R (x ik, x ik  1 )  0, k  1, 2,..., r-1  x ik,  V (x ik )  0, k  1, 2,..., r l(x i1, x i2,..., x ir )   R (x i1, x i2 )   R (x i2, x i3 ) ...   R (x ir  1, x ir )   V (x i1 )   V (x i2 ) ...   V (x ir )

4.2 Characteristics of Fuzzy Relation 4.2.1 Reflexive Relation For all x  A, if  R (x, x)  1 Example 4.8 A  {2, 3, 4, 5} R : For x, y  A, “ x is close to y ” If  x  A,  R (x, y) ≠ 1, then the relation is called “ irreflexive ”. If  x  A,  R (x, y)≠1, then it is called “ antireflexive ” R2345 21.00.90.80.7 30.91.00.90.8 4 0.91.00.9 50.70.80.91.0

4.2.2 Symmetric Relation Symmetric  (x, y)  A  A  R (x, y)     R (y, x) =  Antisymmetric  (x, y)  A  A, x  y  R (x, y)   R (y, x) or  R (x, y)   R (y, x)  0 asymmetric or nonsymmetic  (x, y)  A  A, x  y  R (x, y)   R (y, x) Perfect antisymmetric  (x, y)  A  A, x  y  R (x, y)  0   R (y, x)  0

4.2.3 Transitive Relation Definition  (x, y), (y, x), (x, z)  A  A  R (x, z)  Max[Min(  R (x, y),  R (y, z))] If we use the symbol  for Max and  for Min, the last condition becomes  R (x, z)   [  R (x, y)   R (y, z)] If the fuzzy relation R is represented by fuzzy matrix M R, we know that left side in the above formula corresponds to M R and right one to M R2. That is, the right side is identical to the composition of relation R itself. So the previous condition becomes, M R  M R2 or R  R 2

4.2.3 Transitive Relation Transitive relation example 0.2 1 0.4 1 0.3 0.6 a c b Fig 4.20 Fuzzy relation (transitive relation) For (a, a), we have  R (a, a)   R2 (a, a) For (a, b),  R (a, b)   R2 (a, b) We see M R  M R2 or R  R 2

4.2.4 Transitive Closure fuzzy matrix M R2 corresponding composition R 2 shall be calculated by the multiplication of M R  R2 (x, z)  M R  M R  Max [Min(  R (x, y),  R (y, z))] Transitive relation was referred to as R  R 2 and thus the relation between M R and M R2 holds M R  M R2 From the property of closure, the transitive closure of R shall be R  = R  R 2  R 3   Generally, if we go on multiplying fuzzy matrices(i.e, composition of relation), the following equation is held. R k = R k+1, k  n So, R  is easily obtained R  = R  R 2  R 3    R k, k  n

4.2.4 Transitive Closure (Example) abc A0.21.00.4 B0.00.60.3 C0.01.00.3 abc A0.20.60.3 B0.00.60.3 C0.00.60.3 abc A0.20.60.3 B0.00.60.3 C0.00.60.3

4.2.4 Transitive Closure (Example) abc A0.21.00.4 B0.00.60.3 C0.01.00.3 abc A0.20.60.3 B0.00.60.3 C0.00.60.3 abC A0.21.00.4 B0.00.60.3 C0.01.00.3

4.3 Classification of Fuzzy Relation 4.3.1 Fuzzy Equivalence Relation Definition(Fuzzy equivalence relation) (1) Reflexive relation  x  A   R (x, x)  1 (2) Symmetric relation  (x, y)  A  A,  R (x, y)     R (y, x)   (3) Transitive relation  (x, y), (y, z), (x, z)  A  A  R (x, z) ≥ Max[Min[  R (x, y),  R (y, z)]] y

4.3 Classification of Fuzzy Relation Example 4.12 (Graph of fuzzy equivalence relation ) a1.00.80.71.0 b0.81.00.70.8 c0.7 1.00.7 d1.00.80.71.0 cbad

4.3 Classification of Fuzzy Relation Application1 : Partition of sets set A is done partition into subsets A 1, A 2,.... by the equivalence relation Example 4.13 a1.00.51.00.0 b0.51.00.50.0 c1.00.51.00.0 d 1.00.5 e0.0 0.51.0 abcde

4.3 Classification of Fuzzy Relation Application 2 : Partition by  -cut  -cut equivalence relation R   R (x, y) = 1 if  R (x, y)  ,  x, y  A i = 0 otherwise

4.3 Classification of Fuzzy Relation Example 4.13  (A/R 0.5 )  {{a, b}, {d}, {c, e, f}} a1.00.80.00.40.0 b0.81.00.00.40.0 c 1.00.01.00.5 d0.4 0.01.00.0 e 1.00.01.00.5 f0.0 0.50.00.51.0 abcdef

4.3 Classification of Fuzzy Relation Application 3 : Set similar to element x If similarity relation R is defined on set A, elements related to arbitrary member x  A can make up "set similar to x". Certainly this set shall be fuzzy one Example 4.14 similarity class of d. : {(a, 0.4), (b, 0.4), (d, 1)} elements similar to element e : {(c, 1), (e, 1), (f, 0.5)}

4.3 Classification of Fuzzy Relation 4.3.2 Fuzzy Compatibility Relation Definition(Fuzzy compatibility relation) 1) Reflexive relation x  A   R (x, x)  1 2) Symmetric relation  (x, y)  A  A  R (x, y)     R (y, x)    -cut to the fuzzy compatibility relation if  R (x, y)    x, y  A i = 0 otherwise

4.3 Classification of Fuzzy Relation Example 4.15 f a1.00.80.0 b0.81.00.0 c 1.0 0.80.0 d 1.0 0.80.7 e0.0 0.8 1.00.7 0.0 0.7 1.0 abcdef f Compatibility relation graph Compatibility covering tree

4.3 Classification of Fuzzy Relation 4.3.3 Fuzzy Pre-order Relation Definition(Fuzzy pre-order relation) (1) Reflexive relation  x  A   R (x, x)  1 (2) Transitive relation  (x, y), (y, z), (x, z)  A  A  R (x, z)  Max[Min(  R (x, y),  R (y, z))] y

4.3 Classification of Fuzzy Relation Example 4.16 (semi-pre-order relation ) transitive but not reflexive a0.21.00.4 b0.00.6 0.3 c0.01.00.3 abc

4.3 Classification of Fuzzy Relation Example 4.17 (anti-reflexive fuzzy pre-order relation ) relation  R (x, x)  0 for all x, a0.0 1.0 0.4 b0.0 0.3 c0.0 1.0 0.0 abc

4.3 Classification of Fuzzy Relation 4.3.4 Fuzzy Order Relation Definition(Fuzzy order relation) (1) Reflexive relation  x  A   R (x, x )  1 (2) Antisymmetric relation  (x, y)  A  A  R (x, y)   R (y, x) or  R (x, y)   R (y, x)  0 (3) Transitive relation  (x, y), (y, z), (x, z)  A  A  R (x, z)  Max[Min(  R (x, y),  R (y, z))] y

4.3 Classification of Fuzzy Relation Example 4.18 (fuzzy order relation ) 0.3 1 0.4 0.8 0.2 0.1 a c b d 11 1

4.3 Classification of Fuzzy Relation Definition(Corresponding crisp order) i) if  R (x, y)   R (y, x)then ii) if  R (x, y)   R (y, x)then

4.3 Classification of Fuzzy Relation Example 4.19 a1100 b0100 c1111 d0001 abcd Crisp order relation obtained from fuzzy order relation)

4.3 Classification of Fuzzy Relation Definition(Dominating and dominated class) R (x, y)  0, Say that x dominates y and denote x  y. 1) The one is dominating class of element x. Dominating class R  [x] which dominates x is defined as,  R  [x] (y)   R (y, x) 2) The other is dominated class. Dominated class R  [x] with elements dominated by x is defined as,  R  [x] (y)   R (x, y)

4.3 Classification of Fuzzy Relation Example 4.20 Dominating class of element a and b R  [a]  {(a, 1.0), (b, 0.7), (d, 1.0)} R  [b]  {(b, 1.0), (d, 0.9)} dominated class by a R  [a]  {(a, 1.0), (c, 0.5)} a1.00.00.50.0 b0.71.00.70.0 c 1.00.0 d1.00.91.0 abcd

4.3 Classification of Fuzzy Relation fuzzy upper bound of subset = {x, y} Example 4.21 fuzzy upper bound = {a, b} R  [a]  R  [b] = {(a, 1.0), (b, 0.7), (d, 1.0)}  {(b, 1.0), (d, 0.9)}  {(b, 0.7), (d, 0.9)}

4.4.1 Fuzzy Ordinal Relation Definition(Fuzzy ordinal relation) (1) Reflexive relation x  A   R (x, x)  1 (2) Antisymmetric relation  (x, y)  A  A  R (x, y)   R (y, x) or  R (x, y)   R (y, x)  0 (3) i) if  R (x, y) >  R (y, x)then ii) if  R (x, y)   R (y, x)then

4.4.1 Fuzzy Ordinal Relation Example 4.22 a b c d e a 1.0 0.9 0.0 0.8 b 0.4 1.0 0.5 1.0 0.3 c 0.9 0.8 1.0 0.0 d 1.0 0.0 e 0.2 0.0 1.0

4.4.1 Fuzzy Ordinal Relation Example 4.22 a b c d e a 1 1 0 0 1 b 0 1 0 1 1 c 1 1 1 0 0 d 0 0 0 1 0 e 0 0 0 0 1

4.4.1 Fuzzy Ordinal Relation Example 4.23 f(d)  f(e)  0 f(b)  1 f(a)  2 f(c)  3

4.4.2 Dissimilitude Relation Definition(Dissimilitude relation)  Antiflexive relation x  A   R (x, x)  0  Symmetric relation for  (x, y)  A  A,  R (x, y) =  R (y, x)  Min - Max transitive relation for  (x, y), (y, z), (x, z)  A  A,  R (x, z)   [  R (x, y)   R (y, z)] y

4.4.2 Dissimilitude Relation Example 4.24 for  (a, b)  0.2, check paths with length 2  (a, a)   (a, b)  0.0  0.2  0.2  (a, b)   (b, b)  0.2  0.0  0.2  (a, c)   (c, b)  0.3  0.3  0.3  (a, d)   (d, b)  0.0  0.2  0.2 The minimum of these values is 0.2  (a, b)  0.2 a b c d a 0.0 0.2 0.3 0.0 b 0.2 0.0 0.3 0.2 c 0.3 0.0 0.3 d 0.0 0.2 0.3 0.0

4.4.3 Fuzzy Morphism Homomorphism R  A  A, S  B  B homomorphism function h : A  B from (A, R) to (B, S) For x 1, x 2  A (x 1, x 2 )  R  (h(x 1 ), h(x 2 ))  S “ If two elements x 1 and x 2 are related by R, their images h(x 1 ) and h(x 2 ) are also related by S ”

4.4.3 Fuzzy Morphism Strong homomorphism R  A  A, S  B  B h : A  B For all x 1, x 2  A, (x 1, x 2 )  R  (h(x 1 ), h(x 2 ))  S For all y 1, y 2  B, if x 1  h  1 (y 1 ), x 2  h  1 (y 2 ) then (y 1, y 2 )  S  (x 1, x 2 )  R

4.4.3 Fuzzy Morphism Fuzzy homomorphism Fuzzy relation R  A  A, S  B  B Function h : A  B Satisfies For all x 1, x 2  A  R (x 1, x 2 )   S [h(x 1 ), h(x 2 )] “ The strength of the relation S for (h(x 1 ), h(x 2 )) is stronger than or equal to the that of R for (x 1, x 2 ). ”

4.4.3 Fuzzy Morphism Fuzzy strong homomorphism Fuzzy relation R  A  A, S  B  B function h : A  B satisfies For all x j  A j, x k  A k, A j, A k  A y 1 =h(x j ), y 2 = h(x k ) y 1, y 2  B, (y 1, y 2 )  S, Max  R (x j, x k ) =  S (y 1, y 2 ) xj, xk

4.4.4 Examples of Fuzzy Morphism Example 4.25 all (x 1, x 2 )  R A has the relation (h(x 1 ), h(x 2 ))  S in B  R (x 1, x 2 )   S (h(x 1 ), h(x 2 )) h(c) = , h(d) = ,  R (c, d) = 0   S ( ,  ) = 0.6 h :a, b   c   d  

4.4.4 Examples of Fuzzy Morphism Example 4.25

4.4.4 Examples of Fuzzy Morphism Example 4.26 Examples of Fuzzy Strong Morphism ( ,  )  S,  S ( ,  ) = 1, h  1 (  ) = {b, c}, h  1 (  ) = {d, e} Max [  R (c, d),  R (c, e)] = Max [ 1, 0.5] = 1 =  S ( ,  ) h :a   b, c   d, e  

4.4.4 Examples of Fuzzy Morphism Example 4.26

Chapter 4 Fuzzy Graph and Relation. 4.1.1 Graph and Fuzzy Graph Graph G  (V, E) V : Set of vertices(node or element) E : Set of edges An edge is pair.

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Chapter 4 Fuzzy Graph and Relation. 4.1.1 Graph and Fuzzy Graph Graph G  (V, E) V : Set of vertices(node or element) E : Set of edges An edge is pair.

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Presentation on theme: "Chapter 4 Fuzzy Graph and Relation. 4.1.1 Graph and Fuzzy Graph Graph G  (V, E) V : Set of vertices(node or element) E : Set of edges An edge is pair."— Presentation transcript:

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