Part of Lecture series on EE646, Fuzzy Theory & Applications delivered by me during First Semester of M.Tech. Instrumentation & Control, 2012 Z H College of Engg. & Technology, Aligarh Muslim University, Aligarh Reference Books: 1. T. J. Ross, "Fuzzy Logic with Engineering Applications", 2/e, John Wiley & Sons,England, 2004. 2. Lee, K. H., "First Course on Fuzzy Theory & Applications", Springer-Verlag,Berlin, Heidelberg, 2005. 3. D. Driankov, H. Hellendoorn, M. Reinfrank, "An Introduction to Fuzzy Control", Narosa, 2012. Please comment and feel free to ask anything related. Thanks!

1. EE-646
Lecture-2

2. Binomial Operations
on
Fuzzy Sets
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3. Norm Operators
• T-Norm Operator (T stands for triangular)
•  is taken as min. (generally), i.e.
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 ( ) ( ) ( ),A B A BA B x x x x X       
( ) ( ) min ( ), ( )A B A Bx x x x       

4. T- Norm Operator
• However,  can also be defined as:
i. Algebraic Product, •
ii. Bounded Product,
iii. Drastic Product,
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
or
 
 

5. i. Algebraic Product
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     ,A B A BA B x x x x X       

6. ii. Bounded Product
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 
 max 0, ( ) ( ) 1 ,
A B
A B
A B x
x x x X

 
  
    
 

7. iii. Drastic Product
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( ), if ( ) 0
( ) ( ), if ( ) 0
0, otherwise
B A
A B A B
A B
x x
x x x
 
  






 



8. Note That
A B A B A B A B

        
Illustration on Board
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9. S-Norm Operator
• S-Norm Operator
•  is taken as max. (generally), i.e.
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 ( ) ( ) ( ),A B A BA B x x x x X       
( ) ( ) max ( ), ( )A B A Bx x x x       

10. S- Norm Operator
• However,  can also be defined as:
i. Algebraic Sum, +
ii. Bounded Sum,
iii. Drastic Sum,
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
or



11. i. Algebraic Sum
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( ) ( ) ( ) ( ) ( ),A B A B A B
A B
x x x x x x X    

     

12. ii. Bounded Sum
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  ( ) min ( ) ( ) , 1 ,A B A B
A B
x x x x X  

   

13. iii. Drastic Sum
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( ), if ( ) 0
( ) ( ), if ( ) 0
1, otherwise
B A
A B A B
A B
x x
x x x
 
  






 



14. Note That
A B A B A B A B X

       
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15. Some More Operations
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16. 1. Difference
 
  
or |
( ) max ( ) ( ) , 0A B A B
A B A B
x x x
x X
  

 
 
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17. 2. Absolute Difference
( ) ( ) ( ) ,A B A B
A B
x x x x X  

   
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18. 3. λ - Sum
 ( ) λ ( ) 1 λ ( ) ,
and λ [0, 1]
A B A B
A B
x x x
x X
      
  
λ
λ
+
+
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19. Order of Magnitude
A B A B A B
A B A B A B X

       
    λ+
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20. Problem Task
Consider two fuzzy sets A and B as below:
Find
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0 0.5 0.3 0.7 0.9 0.2 0.4 0.6 0.9 0.4
+ + + + & + + + +
1 2 3 4 5 1 2 3 4 5
A B
   
    
   
(a) , (b) , (c) &
(d) | , (e)Verify De'Morgans Laws
(f) Algebraic Sum, and (g) Algebraic Product
C C
A B A B A B
A B
 

21. Announcement
• From tomorrow onwards Thursday classes will
be held in the first period
• Kindly make arrangements for the projector in
advance
• Thank You!
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