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CHAPTER 2 Fuzzy Mathematics

CHAPTER 2 Fuzzy Mathematics. 2.5 Fuzzy Matrix. 2.6 Fuzzy Relation. 2.5 Fuzzy Matrix. 2.5.1 Basic definitions of fuzzy matrix. 2.5.2 Support of fuzzy matrix. 2.5.3 Composition of fuzzy matrix. 2.5.4 Transpose of fuzzy matrix.

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CHAPTER 2 Fuzzy Mathematics

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  1. CHAPTER 2 Fuzzy Mathematics 2.5 Fuzzy Matrix 2.6 Fuzzy Relation

  2. 2.5 Fuzzy Matrix 2.5.1 Basic definitions of fuzzy matrix 2.5.2 Support of fuzzy matrix 2.5.3 Composition of fuzzy matrix 2.5.4 Transpose of fuzzy matrix

  3. If ( , ), there exist , then is a fuzzy matrix. Usually, means all the fuzzy matrixes ( rows, lines). 2 Conjunction: 2.5.1 Basic definitions of fuzzy matrix 1 Disjunction:

  4. 4 Contain: 3 Complement: 5 Equal: 2.5.1 Basic definitions of fuzzy matrix

  5. Example 2.5.1: Example 2.5.2: 2.5.1 Basic definitions of fuzzy matrix Notes: A fuzzy matrix and its complement are not complementary events similar with fuzzy sets.

  6. 1 nilpotent 2 commutative 3 associative 4distributive

  7. 7 8 5 absorptive 6identical 9 DeMorgan’s law

  8. Example 2.5.3: 2.5.1 Basic definitions of fuzzy matrix

  9. Definition: is called the support matrix of . Notes: The support matrix is a Boolean matrix. 2.5.2 Support of fuzzy matrix

  10. 2.5.2 Support of fuzzy matrix Example 2.5.4:

  11. property： 1. 2. 3. 4.

  12. Definition: , , , , the composition matrix of these two matrixes is a fuzzy matrix with rows and lines, written as whose element is The composition of fuzzy matrix is also called as multiplication of fuzzy matrix. 2.5.3 Composition of fuzzy matrix

  13. Example 2.5.5: Notes: (popularly) 2.5.3 Composition of fuzzy matrix

  14. Definition: Assume call is transpose of fuzzy matrix A,where 。 2.5.4 Transpose of fuzzy matrix Rows change into lines and lines change into rows.

  15. property：

  16. 特殊的模糊矩阵 则称A为自反矩阵。 定义：若模糊方阵满足 是模糊自反矩阵。 例如 定义：若模糊方阵满足 则称A为对称矩阵。 例如 是模糊对称矩阵。

  17. 定义：若模糊方阵满足 则称A为模糊传递矩阵。 例如 是模糊传递矩阵。

  18. 2.6 Fuzzy Relation 2.6.1 Definition of fuzzy relation 2.6.2 Operation of fuzzy relation 2.6.3 Composition of fuzzy relation

  19. Relation: A subset of a Cartesian product of and , is called a dualistic relation from to (or relation for short). Fuzzy relation: and are two sets, is a fuzzy subset in , is also called a fuzzy relation from to . The degree of membership of is 2.6.1 Definition of fuzzy relation

  20. 2.6.1 Definition of fuzzy relation Fuzzy relation can described by fuzzy matrix when and are finite set. When , it is called the fuzzy relation in . An element in fuzzy matrix means the relation between the element in and the element in .

  21. is the space of height in an area and is the space of weight. ,(cm), ,(kg), the relation between height and weight is in table. 2.6.1 Definition of fuzzy relation Example 2.5.6

  22. 40 50 60 70 80 140 1 0.8 0.2 0.1 0 150 0.8 1 0.8 0.2 0.1 160 0.2 0.8 1 0.8 0.2 170 0.1 0.2 0.8 1 0.8 180 0 0.1 0.2 0.8 1 2.6.1 Definition of fuzzy relation Fuzzy relation between height and weight

  23. 2.6.1 Definition of fuzzy relation We can find from the above example that how does fuzzy mathematics describe the fuzzy conception in real world in the form of math. Using this method, we can handle the fuzzy conception in computer.

  24. 3 Contain: 4 Equal: 5 Complement: 6 Transpose: 2.6.2 Operation of fuzzy relation and are two fuzzy relations in . 1 Union: 2 Intersection:

  25. 2.6.2 Operation of fuzzy relation Characteristics of fuzzy relation: 1 自反性 Every element in diagonal of fuzzy matrix is 1. 2 对称性 3 传递性

  26. 2.6.2 Operation of fuzzy relation Example 2.6.1 Two people have the fuzzy relations: “similitude” 自反性（Y）对称性（Y）传递性（N） “enemy” 自反性（N）对称性（Y）传递性（N） “love” 自反性（Y）对称性（?）传递性（?） “younger” 自反性（N）对称性（N）传递性（Y）

  27. 2.6.2 Operation of fuzzy relation Example 2.6.2 自反性（Y）对称性（Y）传递性（N）

  28. is a space of people, is the brother relation, is the paternity, is the relation between uncle and nephew. , , are three people in , is a brother of , is the father of , so should be the uncle of . We call the relation between uncle and nephew is the composition between the brother relation and paternity. And written as： If then 2.6.3 Composition of fuzzy relation

  29. In a general way, ， ， are the spaces, is a common relation from to , is a common relation from to , and is a common relation form to , that is, if then is called a composition from to , written as : 2.6.3 Composition of fuzzy relation

  30. Definition: Let ， ， be the spaces, if is a fuzzy relation from to , is a fuzzy relation from to , and is a fuzzy relation form to , then is the composition from to which has the membership function: When ， ， are finite, the composition of fuzzy relation could be expressed by the composition of fuzzy matrices. 2.6.3 Composition of fuzzy relation

  31. Emphases 1 Operation of fuzzy matrix. 2 Characteristics of fuzzy relation.

  32. Words 1 模糊矩阵 2 模糊矩阵的合成 3 模糊关系 fuzzy matrix composition of fuzzy matrices fuzzy relation

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