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

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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

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