The document outlines the fundamental concepts of fuzzy set theory compared to traditional crisp set theory, emphasizing the difference in membership functions. In fuzzy sets, elements can have partial membership represented on a scale from 0 to 1, allowing for vagueness in definitions and boundaries. The document also discusses operations on fuzzy sets, such as subset, complement, intersection, union, and various properties associated with these operations.

1. EE-646
Fuzzy Theory & Applications
Lecture-1

2. Crisp Sets
• Let X denotes the Universe of Discourse, whose
generic elements are denoted by x.
• Membership function or characteristic function
µA(x) in crisp set maps whole members in
universal set X to set {0, 1}.
• µA(x): X → {0, 1}
• “well- defined” boundary
• No partial membership allowed
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3. Fuzzy Sets
• In fuzzy sets, each elements is mapped to [0, 1] by
membership function:
• µA(x) : X → [0, 1]
• “Vague” boundary
• Partial membership is allowed
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4. Conventional (Boolean) Set Theory
Crisp Vs. Fuzzy
“Strong Fever”
40.1°C
42°C
41.4°C
39.3°C
38.7°C
37.2°C
38°C
Fuzzy Set Theory:
40.1°C
42°C
41.4°C
39.3°C
38.7°C
37.2°C
38°C
“More-or-Less” Rather
Than “Either-Or” ! “Strong Fever”

5. Another Example
Seasons
Discuss yourselves on age, temperature, height as
Fuzzy Sets (Homework)
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6. Fuzzy Set Representation
• Ordered pair of an element and the
corresponding membership value.
• Discrete Case:
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  , ( ) : AA x x x A
     1 2
1 2
...A i A A
i i
x x x
A
x x x
  
   
Not Addition!

7. Fuzzy Set Representation...contd
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Continuous case:
Not to be confused with integration!!
Graphical Representation: See Board
    1 1 2 2, ( ) , , ( ) ,...A AA x x x x 
 A i
ii
x
A
x

 

8. Operations on Fuzzy Sets
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9. 1. Subset
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( ) ( ),A B
A B
x x x X 

  
A is contained in B
Graph on Board

10. 2. Complement
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or '
( ) 1 ( ),
C
B A
B A A
x x x X 

   

11. 3. Intersection
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 ( ) ( ) ( ),A B A Bx x x x X      
T-norm operator
Can be defined in a no. of ways

12. 4. Union
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 ( ) ( ) ( ),A B A Bx x x x X      
T-Conorm operator
Or S-Norm operator

13. 5. Law of Excluded Middle
These laws are not valid in case of Fuzzy Sets!
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6. Law of Contradiction
Rather,
C
C
A A U
A A U
 
 
Rather,
C
C
A A
A A


 
 

14. 7. Idempotency
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&A A A A A A   
8. Commutativity
&A B B A A B B A     

15. 9. Associativity
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   
   
A B C A B C
A B C A B C
    
    
10. Absorption
   A A B A A B A     

16. 11. Distribution
Write yourself (Right Now!)
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12. Double Negation
13. De’ Morgans Laws
     
     
A B C A B A C
A B C A B A C
     
     
 
CC
A A
   and
C CC C C C
A B A B A B A B     

17. Task
Verify all these properties graphically
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18. Power of a Fuzzy Set
• mth power of Fuzzy Set A is denoted by Am
• Defined as
• This operator will be used later to model
linguistic hedges
• Illustration on Board
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( ) ( ) ,m
m
AA
x x x X     