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Chapter 2 Fuzzy Sets Versus Crisp Sets

Chapter 2 Fuzzy Sets Versus Crisp Sets. Part one: Theory. 2.1 Additional properties of alpha-cuts. Alpha-cuts and strong alpha-cuts are always monotonic decreasing with respect to alpha. The standard fuzzy intersection and fuzzy union are both cutworthy when applied to two fuzzy sets.

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Chapter 2 Fuzzy Sets Versus Crisp Sets

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  1. Chapter 2Fuzzy Sets Versus Crisp Sets Part one: Theory

  2. 2.1 Additional properties of alpha-cuts Alpha-cuts and strong alpha-cuts are always monotonic decreasing with respect to alpha The standard fuzzy intersection and fuzzy union are both cutworthy when applied to two fuzzy sets. The standard fuzzy intersection and fuzzy union are both strong cutworthy when applied to two fuzzy sets. The standard fuzzy complement is neither cutworthy nor strong cutworthy.

  3. 2.1 Additional properties of alpha-cuts P. 23

  4. 2.1 Additional properties of alpha-cuts

  5. 2.1 Additional properties of alpha-cuts

  6. 2.1 Additional properties of alpha-cuts

  7. 2.1 Additional properties of alpha-cuts

  8. 2.1 Additional properties of alpha-cuts • An example: (vi) (a)

  9. 2.1 Additional properties of alpha-cuts

  10. 2.1 Additional properties of alpha-cuts

  11. 2.2 Representations of fuzzy sets • In this section, we show that each fuzzy set can uniquely be represented by either the family of all its -cuts or the family of all its strong -cuts. • Representations of fuzzy sets by crisp sets (the first one): • An example: Considering the fuzzy set this can be represented by its -cuts:

  12. 2.2 Representations of fuzzy sets • Define a fuzzy set we obtain Now, it is easy to see that

  13. 2.2 Representations of fuzzy sets

  14. 2.2 Representations of fuzzy sets • For example:

  15. 2.2 Representations of fuzzy sets

  16. 2.2 Representations of fuzzy sets

  17. 2.2 Representations of fuzzy sets • For example: • The level set of A: and

  18. 2.3 Extension principle for fuzzy set • A crisp function: f : X  Y • A fuzzified function • Its inverse function • An extension principle: • a principle for fuzzifying crisp functions • Now, we first discuss the extended functions which are restricted to crisp power sets.

  19. B(y) = A(x) = X Y P(X) P(Y) f f x y y x A B 2.3 Extension principle for fuzzy set

  20. 2.3 Extension principle for fuzzy set • An example: • Let X={a, b, c} and Y={1,2}

  21. B(y) = A(x) = 2.3 Extension principle for fuzzy set

  22. 0.8 0.4 0.2 0.4 0.7 0.8 2.3 Extension principle for fuzzy set

  23. 2.3 Extension principle for fuzzy set

  24. 2.3 Extension principle for fuzzy set

  25. 2.3 Extension principle for fuzzy set

  26. 2.3 Extension principle for fuzzy set

  27. 2.3 Extension principle for fuzzy set

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