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Chapter 4 Fuzzy Graph and Fuzzy relation

Chapter 4 Fuzzy Graph and Fuzzy 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.

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Chapter 4 Fuzzy Graph and Fuzzy relation

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  1. Chapter 4Fuzzy Graph and Fuzzy relation

  2. 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

  3. 4.1.2 Fuzzy graph and fuzzy relation • Example 4.1 (an example of fuzzy graph) Fig 4.1 Fuzzy graph

  4. 4.1.2 Fuzzy graph and fuzzy relation • Example 4.2 •  : nonnegative real numbers. • x and y • R {(x,y) | xy}, R . Fig 4.1 Fuzzy graph yx ( y closes to x)

  5. 4.1.2 Fuzzy graph and fuzzy relation • 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 Fig 4.3 Fuzzy relation and fuzzy graph

  6. 4.1.2 Fuzzy graph and fuzzy relation • Example 4.4 mapping function (A) {a1}  {(b1, 0.5), (b2, 1.0)} {a2}  {(b3, 0.5)} {a3}  {(b1, 1.0), (b2, 1.0)} {a1, a2}  {(b1, 0.5), (b2, 1.0), (b3, 0.5)} Fig 4.4 Fuzzy graph

  7. 4.1.2 Fuzzy graph and fuzzy relation • Example 4.5 A picture and fuzzy relation Fig 4.5 Fuzzy graph Fig 4.6 Fuzzy graph(by coordinates)

  8. y 1 1 x 1 -1 y 1 1 x -1 -1 4.1.2 Fuzzy graph and fuzzy relation • Example 4.6 A graph and a fuzzy graph (b) Graph R (x,y) x2 + y2 1 (a) Graph R(x,y) x2 + y2 1 Fig 4.7 Fuzzy graph

  9. a c b 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.

  10. 4.1.3 -cut of Fuzzy Graph

  11. 4.1.3 -cut of Fuzzy Graph • Example 4.8 R(x, y) = x/2 + y 1 Fig 4.9 Graphical form of R Fig 4.10 Graphical representation of R0.5

  12. 4.1.3 -cut of Fuzzy Graph • Example 4.9 A(x) = x R(x,y) = x+y  1, x  A, 0  y  1 Fig 4.11 Set A(x)= x Fig 4.12 Relation A(x,y)= x+y  1 , xA

  13. A(x) x 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]. Fig 4.14 -cut set A0.5 Fig 4.13 Set A(x)=cosx 0

  14. 4.1.3 -cut of Fuzzy Graph Fig 4.15 Relation R(x,y)=cosx. Fig 4.16 -cut relation R0.5

  15. 4.2 Characteristics of Fuzzy Relation • 4.2.1 Reflexive Relation • For all xA, if R(x, x)  1 • Example 4.8A  {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 ”

  16. 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

  17. 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 MR, we know that left side in the above formula corresponds to MR and right one to MR2. That is, the right side is identical to the composition of relation R itself. So the previous condition becomes, • MRMR2 or R R2

  18. 0.6 b 1 0.2 a 1 0.3 0.4 c 0.3 4.2.3 Transitive Relation • Transitive relation example For (a, a), we have R(a, a) R2(a, a) For (a, b), R(a, b) R2(a, b) We see MRMR2 or R R2 Fig 4.20 Fuzzy relation (transitive relation)

  19. 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

  20. a b c d a 1.0 0.8 0.7 1.0 b 0.8 1.0 0.7 0.8 c 0.7 0.7 1.0 0.7 d 1.0 0.8 0.7 1.0 4.3 Classification of Fuzzy Relation • Example 4.12 (Graph of fuzzy equivalence relation )

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