Fuzzy sets Operation

                   FUZZY  SET  OPERATIONS :
1. FUZZY  UNION :-
         A = {( x , 0.5) , (y , 0.7) , (z ,0)}  

          B={( x , 0.8) , (y , 0.2) , (z ,1)}
A U B = A UNION B
         = maximum of membership       
     degree among the sets
A U B = ( x , 0.8) , (y , 0.7) , (z ,1)}
 NOTICE -
          0.8 > 0.5
           0.7 > 0.2
            1 > 0

2. FUZZY  INTERSECTION :-
 A = {( x , 0.5) , (y , 0.7) , (z ,0)}         B={( x , 0.8) , (y , 0.2) , (z ,1)}
 A INTERSECTION B  = minimum of
membership degree among the sets
= ( x , 0.5) , (y , 0.2) , (z ,0)}

 NOTICE - 
          0.8 < 0.5
           0.7 < 0.2
            1 < 0

3.  A ' = COMPLEMENT OF A
A = {( x , 0.5) , (y , 0.7) , (z ,0)}         B={( x , 0.8) , (y , 0.2) , (z ,1)}
A ' = 1 - MEMBERSHIP DEGREE
A ' = {(x , 0.5), (y ,0.3) , (z,1)}
B ' ={(x , 0.2) , (y,0.8) , (z,0)}

4.  A - B = ? 
A = {( x , 0.5) , (y , 0.7) , (z ,0)}        
B={( x , 0.8) , (y , 0.2) , (z ,1)}
A - B = A INTERSECTION B'
         = minimum of membership       
     degree among the sets
A- B = {(x , 0.2) , (y , 0.7) , (z ,0)}

5. POWER OF  FUZZY SET = ? 
   IF "A " IS A FUZZY SET THEN
     POWER SET (A)  =  (membership
degree of set )^ $
       P(A) = (M. DEGREE)^$
NOTE -
        IF $ = 1/2 THEN IT IS CALLED
"DILATION"
       IF $ = 2 THEN IT IS CALLED
"CONCENTRATION "

EXAMPLE - 
A = {( x , 0.5) , (y , 0.7) , (z ,0)} 
$ = 2
A = {( x , 0.25) , (y , 0.49) , (z ,0)} 

Play with fuzzy sets

 PLAY WITH  FUZZY  SETS :

A set or a crisp set is defined  as a 

collection of elements . The elements are 

either a member or not a member of the 

set . But Fuzzy sets are more flexible in 

approach  as compared  to the sets .

Fuzzy set can be viewed as an 

extension & generalization 0f  the 

basic concepts of crisp set .It supports 

flexible sense of membership with a 

parameter called "membership  degree" . 

It is multivalued between 0 and 1 (both 

included). You  will be  surprised  to 

know that Fuzzy logics donot follow 

LAW  OF EXCLUDED MIDDLE & LAW 

OF CONTRADICTION .If  we will choose 

fuzzy union & intersection that will 

follow these law then , the operations 

will not satisfy distribution and 

idempotence .

LAW  OF EXCLUDED MIDDLE states 

that for any proposition , either that 

proposition is true , or its negation is 

true .
LAW  OF CONTRADICTION  states that 

two antiethical propositions cannot be 

true at the same time . For example - X 

cannot be NOT X simultaneously.
Distributive property :-
                        5(x+30) = 5x + 150 
Idempotence property :-
                          op( op(x)) == op(x)