CHAPTER 2 Fuzzy Mathematics 2 5 Fuzzy Matrix

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

2. 5. 1 Basic definitions of fuzzy matrix If ( , ), there exist , then is a fuzzy matrix. Usually, means all the fuzzy matrixes ( rows, lines). 1 Disjunction: 2 Conjunction:

2. 5. 1 Basic definitions of fuzzy matrix 3 Complement: 4 Contain: 5 Equal:

2. 5. 1 Basic definitions of fuzzy matrix Example 2. 5. 1: Notes: A fuzzy matrix and its complement are not complementary events similar with fuzzy sets. Example 2. 5. 2:

1 nilpotent 2 commutative 3 associative 4 distributive

5 absorptive 6 identical 7 8 9 De. Morgan’s law

2. 5. 1 Basic definitions of fuzzy matrix Example 2. 5. 3:

2. 5. 2 Support of fuzzy matrix Definition: is called the support matrix of Notes: The support matrix. . is a Boolean

2. 5. 2 Support of fuzzy matrix Example 2. 5. 4:

property： 1. 2. 3. 4.

2. 5. 3 Composition of fuzzy matrix 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 Example 2. 5. 5: Notes: (popularly)

2. 5. 4 Transpose of fuzzy matrix Rows change into lines and lines change into rows. Definition: Assume call transpose of fuzzy matrix A, where is 。

property ：

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

2. 6. 1 Definition of fuzzy relation 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 Fuzzy relation can described by fuzzy matrix when and are finite set. When , it is called the fuzzy relation in. An element between the in. in fuzzy matrix means the relation element in and the element

2. 6. 1 Definition of fuzzy relation Example 2. 5. 6 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 Fuzzy relation between height and weight 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. 8 180 0 0. 1 0. 2 0. 8 1

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.

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

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

2. 6. 2 Operation of fuzzy relation Example 2. 6. 2 自反性（Y）对称性（Y）传递性（N）

2. 6. 3 Composition of fuzzy relation 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 In a general way, ， ， are the spaces, is a common relation from to , and is a common relation form to , that is, if then as : is called a composition from to , written

2. 6. 3 Composition of fuzzy relation Definition: Let ， ， be the spaces, if 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.

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

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