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Objectives

1) To understand fuzzy numbers Using alpha-cut representation and the extension principle.

2) To know the different between crisp relations and fuzzy relations.

3) Can give an example to representation fuzzy relations.

4) To know what is composition of fuzzy relation and can give an example of its.

Fuzzy Numbers

There are two fuzzy numbers,

                                      A = 3 = 0.3/1 + 0.7/2 + 1/3 + 0.7/4 + 0.3/5           B = 7 = 0.2/5 + 0.6/6 + 1/7 + 0.6/8 + 0.2/9

can represent in table1 for easy to understand 

 

 

Table1. Fuzzy numbers about3 and about7

Addition

                       Using alpha-cuts representation                                           Using the extension principle

Figure1.1 addtion using alpha-cuts

Figure1.2 addtion using extension principle

          From figure1.1, show the result of addtion fuzzy numbers about3 and about7 is about10 because number10 has degree of membership equal to 1. 

          From figure1.2, the result show that summation of number3 from fuzzy set A and number7 from fuzzy set B is equal to 10 and degree of number 10 is 1.0 so about10 is the result from this addition.

                           

                                     Thus   C = A+B = 0.2/6 + 0.3/7 + 0.6/8 + 0.7/9 + 1.0/10 + 0.7/11 + 0.6/12 + 0.3/13 + 0.2/14

 

 

 

 

Subtraction

          From figure2.1, show the result of subtraction fuzzy numbers about3 and about7 is about -4 because number -4 has degree of membership equal to 1. 

          From figure2.2, the result show that subtraction of number3 from fuzzy set A and number7 from fuzzy set B is equal to -4 and degree of number -4 is 1.0 so about -4 is the result from this subtraction.

 

                                    Thus   C = A-B = 0/-9 + 0.2/-8 + 0.3/-7 + 0.6/-6 + 0.7/-5 + 1/-4 + 0.7/-3 + 0.6/-2 + 0.3/-1 + 0.2/0 + 0/1

 

                       Using alpha-cuts representation                                           Using the extension principle

Figure2.1 subtraction using alpha-cuts

Figure2.2 subtraction using extension principle

Multiplication

                       Using alpha-cuts representation                                           Using the extension principle

Figure3.1 Multiplication using alpha-cuts

Figure3.2 Multiplication using extension principle

          From figure3.1, show the result of multiplication fuzzy numbers about3 and about7 is about 21 because number 21 has degree of membership equal to 1. 

          From figure3.2, the result show that multiplication of number3 from fuzzy set A and number7 from fuzzy set B is equal to 21 and degree of number 21 is 1.0 so about 21 is the result from this multiplication.

 

                                  

 

Division

                       Using alpha-cuts representation                                           Using the extension principle

Figure4.1 Division using alpha-cuts

Figure4.2 Division using extension principle

          From figure4.1, show the result of division fuzzy numbers about3 and about7 is about 0.4 because nearest number 0.4 has degree of membership equal to 1. 

          From figure4.2, the result show that division of number3 from fuzzy set A and number7 from fuzzy set B is nearest to 0.4 and degree of nearest 0.4 is 1.0 so about 0.4 is the result from this dvision.

 

                                  

 

          Relation means to describe the relationship between the data from two or more sets. The crisp relation has degree of association is 0 or 1. But for fuzzy relation has degree of association between 0 and 1.

          Fuzzy relation is a link between 2 or more fuzzy sets will be based on the degree of membership which can be applied to fuzzy clustering, fuzzy control and fuzzy reasoning as well as to evaluate and compare an important of the problem with such a fuzzy diagnosis and fuzzy modeling.

          Fuzzy relation is fuzzy if / then rules by specifying that one thing is the element of another. The order of relationships will be important as well.

 

Example    

          Crisp set X with three linguistic terms is given as 

                     X = {green, yellow, red}

          Similarly the grade of maturity for the other set Y will be

                     Y = {verdant, half-mature, mature}

         Crisp formulation of a relation X to Y between two crisp sets is presented in tabular form

 

 

 

 

 

 

 

 

 

 

          The figure5.2 represents the fuzzy relation.


 

 

 

Fuzzy Relations

Figure5.1 Crisp relations

Figure5.2 Fuzzy relations

Fuzzy Relation Representations

From an example of fuzzy relation in figure5.2 can described in 5 ways

 

1) Tabular form                                                                           2) Matrix  form

3) Direct Graph form

4) Linguistically, “x divides y”

5) By listing the set of all ordered pairs

Projection of Fuzzy Relation 

 

From fuzzy relation R(x,y)                                  

 

 

 

The projection of R(x,y) on X denoted by 

 

 

 

 

and The projection of R(x,y) on Y denoted by 

 

 

 

 

Example

          From previous fuzzy relation

 

 

 

 

 

 

 

          The projection of R(x,y) on X is calculated as

 

 

 

 

 

 

          so the X projection is


 

          The projection of R(x,y) on Y is calculated as

 

 

 

 

 

 

          so the Y projection is


 

 

Cylindrical extension of Fuzzy Relation 

 

          Cylindrical extension from X-projection means filling all the columns of the related matrix by theX -projection.

Similarly cylindrical extension from Y projection means filling all the rows of the relational matrix by the Y -projection.

 

Example

          From the  fuzzy relation

 

 

 

 

 

          The projection of R(x,y) on X is calculated as

 

 

 

 

 

 

          so the X projection is


 

         So The cylindrical extension of R1 is

 

 

 

 

 

 

          The projection of R(x,y) on Y is calculated as

 

 

 

 

 

 

          so the Y projection is


 

          So The cylindrical extension of R2 is

Fuzzy Max-Min Composition

          Let us consider two fuzzy relations R1 and R2 defined on a Cartesian space X x Y and Y x Z respectively. The max-min composition of R1 and R2 is a fuzzy set defined on a cartesian spaces X  x Z as

 

 

 

Example

          Let R1(x,y) and R2(y,z) be defined as the following relational matrices

 

 

 

 

 

 

          the max-min composition can be

 

 

 

 

 

 

          calculated each entries


 

 

 

 

 

 

          The relational matrix for max-min composition in fuzzy relation is thus


 

Fuzzy Max-Product Composition

          Max-Product composition is similar to Max-Min composition but its different at try to multiply between two value of degree of membership instead of to do minimum

 

Example

          Let R1(x,y) and R2(y,z) be defined as the following relational matrices

 

 

 

 

 

 

          the max-product composition can be

 

 

 

 

 

 

          calculated each entries


 

 

 

 

 

 

          The relational matrix for max-product composition in fuzzy relation is thus


 

Fuzzy Max-Average Composition

          Max-Average composition has method like this. First, do the summation of two degree or membership. Then, find the maximum value. Finally, give the average of maximum value divided by 2. 

 

Example

          Let R1(x,y) and R2(y,z) be defined as the following relational matrices

 

 

 

 

 

 

          the max-average composition can be

 

 

 

 

 

 

          calculated each entries


 

 

 

 

 

 

         

 

 

The relational matrix for max-average composition in fuzzy relation is thus


 

Discussion

          From this assignment, i know about fuzzy relation, fuzzy composition and fuzzy number operation. But i'm not sure that my understanding is correct or not because i'm too confuse about it. For fuzzy number operation, i try to coding with MATLAB but it has some error in subtraction and division that i can't solve it so i change to describe it like table by hand addition. Finally, this assignmnet make me feel good with fuzzy because i read a lot and familiar with it and the last, i want to say i'm so sorry about my grammar.

Neungluethai Boorasit

5537857 EGBE/M

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