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

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

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

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

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

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

23. 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

28. 2.3 Extension principle for fuzzy set