L1 fuzzy sets & basic operations
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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.
![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
22 October 2012 3](https://image.slidesharecdn.com/l1-fuzzysetsbasicoperations-150117021739-conversion-gate02/85/L1-fuzzy-sets-basic-operations-3-320.jpg)
L1 fuzzy sets & basic 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 22 October 2012 2
- 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 22 October 2012 3
- 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) 22 October 2012 5
- 6. Fuzzy Set Representation • Ordered pair of an element and the corresponding membership value. • Discrete Case: 22 October 2012 6 , ( ) : 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 22 October 2012 7 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 22 October 2012 8
- 9. 1. Subset 22 October 2012 9 ( ) ( ),A B A B x x x X A is contained in B Graph on Board
- 10. 2. Complement 22 October 2012 10 or ' ( ) 1 ( ), C B A B A A x x x X
- 11. 3. Intersection 22 October 2012 11 ( ) ( ) ( ),A B A Bx x x x X T-norm operator Can be defined in a no. of ways
- 12. 4. Union 22 October 2012 12 ( ) ( ) ( ),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! 22 October 2012 13 6. Law of Contradiction Rather, C C A A U A A U Rather, C C A A A A
- 14. 7. Idempotency 22 October 2012 14 &A A A A A A 8. Commutativity &A B B A A B B A
- 15. 9. Associativity 22 October 2012 15 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!) 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 CC A A and C CC C C C A B A B A B A B
- 17. Task Verify all these properties graphically 22 October 2012 17
- 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 22 October 2012 18 ( ) ( ) ,m m AA x x x X