l1 fuzzy sets & basic operations
TRANSCRIPT
EE-646 Fuzzy Theory & Applications
Lecture-1
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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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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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”
Another Example
Seasons
Discuss yourselves on age, temperature, height as Fuzzy Sets (Homework)
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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 xA
x x x
Not Addition!
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
xA
x
Operations on Fuzzy Sets
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1. Subset
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( ) ( ),A B
A B
x x x X
A is contained in B
Graph on Board
2. Complement
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or '
( ) 1 ( ),
C
B A
B A A
x x x X
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
4. Union
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( ) ( ) ( ),A B A Bx x x x X
T-Conorm operator Or S-Norm operator
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
7. Idempotency
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& A A A A A A
8. Commutativity
& A B B A A B B A
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
11. Distribution
Write yourself (Right Now!) 22 October 2012 16
12. Double Negation
13. De’ Morgans Laws
A B C A B A C
A B C A B A C
C
CA A
and C CC C C C
A B A B A B A B
Task
Verify all these properties graphically
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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
AAx x x X