1. PART 3 Operations on fuzzy sets
FUZZY SETS AND
FUZZY LOGIC
Theory and Applications
PART 3 Operations on fuzzy sets
1. Fuzzy complements
2. Fuzzy intersections
3. Fuzzy unions
4. Combinations of operations
5. Aggregation operations

2. (boundary conditions).
Fuzzy complements
Axiomatic skeleton
Axiom c1.
Axiom c2.
(boundary conditions).
For all if , then (monotonicity).

3. Fuzzy complements Desirable requirements Axiom c3. Axiom c4.
c is a continuous function.
c is involutive, which means that for each

4. Fuzzy complements Theorem 3.1
Let a function satisfy Axioms c2 and c4. Then, c also satisfies Axioms c1 and c3. Moreover, c must be a bijective function.

5. Fuzzy complements

6. Fuzzy complements

7. Fuzzy complements
Sugeno class
Yager class

8. Fuzzy complements

9. Fuzzy complements Theorem 3.2
Every fuzzy complement has at most one equilibrium.

10. Fuzzy complements Theorem 3.3
Assume that a given fuzzy complement c has an equilibrium ec , which by Theorem 3.2 is unique. Then
and

11. Fuzzy complements Theorem 3.4 Theorem 3.5
If c is a continuous fuzzy complement, then c has a unique equilibrium.
If a complement c has an equilibrium ec , then

12. Fuzzy complements

13. Fuzzy complements Theorem 3.6
For each , , that is, when the complement is involutive.

14. Fuzzy complements Theorem 3.7
(First Characterization Theorem of Fuzzy Complements).
Let c be a function from [0, 1] to [0, 1]. Then, c is a fuzzy complement (involutive) iff there exists a continuous function from [0, 1] to R such that , is
strictly increasing, and
for all

15. Fuzzy complements
Increasing generators
Sugeno:
Yager:

16. Fuzzy complements Theorem 3.8
(Second Characterization Theorem of Fuzzy complements).
Let c be a function from [0, 1] to [0, 1]. Then c is a fuzzy complement iff there exists a continuous function from [0, 1] to R such that , is strictly decreasing, and
for all

17. Fuzzy complements
Decreasing generators
Sugeno:
Yager:

18. Fuzzy intersections: t-norms
Axiomatic skeleton
Axiom i1.
Axiom i2.
(boundary condition).
implies (monotonicity).

19. Fuzzy intersections: t-norms
Axiomatic skeleton
Axiom i3.
Axiom i4.
(commutativity).
(associativity).

20. Fuzzy intersections: t-norms
Desirable requirements
Axiom i5
Axiom i6
Axiom i7
is a continuous function (continuity).
(subidempotency).
implies
(strict monotonicity).

21. Fuzzy intersections: t-norms
Archimedean t-norm:
A t-norm satisfies Axiom i5 and i6.
Strict Archimedean t-norm:
Archimedean t-norm and satisfies Axiom i7.

22. Fuzzy intersections: t-norms
Frequently used t-norms

23. Fuzzy intersections: t-norms

24. Fuzzy intersections: t-norms

25. Fuzzy intersections: t-norms
Theorem 3.9
Theorem 3.10
The standard fuzzy intersection is the only idempotent t-norm.
For all ,
where denotes the drastic intersection.

26. Fuzzy intersections: t-norms
Pseudo-inverse of decreasing generator
The pseudo-inverse of a decreasing generator , denoted by , is a function from R to [0, 1] given by
where is the ordinary inverse of .

27. Fuzzy intersections: t-norms
Pseudo-inverse of increasing generator
The pseudo-inverse of a increasing generator , denoted by , is a function from R to [0, 1] given by
where is the ordinary inverse of .

28. Fuzzy intersections: t-norms
Lemma 3.1
Let be a decreasing generator. Then a function defined by
for any is an increasing generator
with , and its pseudo-inverse
is given by
for any R.

29. Fuzzy intersections: t-norms
Lemma 3.2
Let be a increasing generator. Then a function defined by
for any is an decreasing generator
with , and its pseudo-inverse
is given by
for any R.

30. Fuzzy intersections: t-norms
Theorem 3.11 (Characterization Theorem of t-Norms).
Let be a binary operation on the unit interval. Then, is an Archimedean t-norm iff there exists a decreasing generator
such that
for all

31. Fuzzy intersections: t-norms
[Schweizer and Sklar, 1963]

32. Fuzzy intersections: t-norms
[Yager, 1980f]

33. Fuzzy intersections: t-norms
[Frank, 1979]

34. Fuzzy intersections: t-norms
Theorem 3.12
Let denote the class of Yager t-norms.
Then,
for all , where the lower and upper bounds are obtained for and
,respectively.

35. Fuzzy intersections: t-norms
Theorem 3.13
Let be a t-norm and be a
function such that is strictly increasing
and continuous in (0, 1) and
Then, the function defined by
for all ,where denotes the pseudo-inverse of , is also a t-norm.

36. Fuzzy unions: t-conorms
Axiomatic skeleton
Axiom u1.
Axiom u2.

37. Fuzzy unions: t-conorms
Axiomatic skeleton
Axiom u3.
Axiom u4.

38. Fuzzy unions: t-conorms
Desirable requirements
Axiom u5.
Axiom u6.
Axiom u7.

39. Fuzzy unions: t-conorms
Frequently used t-conorms

40. Fuzzy unions: t-conorms

41. Fuzzy unions: t-conorms

42. Fuzzy unions: t-conorms
Theorem 3.14
The standard fuzzy union is the only idempotent t-conorm.

43. Fuzzy unions: t-conorms
Theorem 3.15
For all

44. Fuzzy unions: t-conorms
Theorem 3.16 (Characterization Theorem of t-Conorms).
Let u be a binary operation on the unit interval. Then, u is an Archimedean t-conorm iff there exists an increasing generator such that
for all

45. Fuzzy unions: t-conorms
[Schweizer and Sklar, 1963]

46. Fuzzy unions: t-conorms
[Yager, 1980f]

47. Fuzzy unions: t-conorms
[Frank, 1979]

48. Fuzzy unions: t-conorms
Theorem 3.17
Let uw denote the class of Yager t-conorms.
for all where the lower and upper bounds are obtained for ,
respectively.

49. Fuzzy unions: t-conorms
Theorem 3.18
Let u be a t-conorm and let be
a function such that is strictly increaning
and continuous in (0, 1) and
Then, the function defined by
for all is also a t-conorm.

50. Combinations of operators
Theorem 3.19
The triples
〈min, max, c〉and〈imin, umax, c〉are dual
with respect to any fuzzy complement c.

51. Combinations of operators
Theorem 3.20
Given a t-norm i and an involutive fuzzy complement c, the binary operation u on [0, 1] defined by
for all is a t-conorm such that
〈i, u, c〉is a dual triple.

52. Combinations of operators
Theorem 3.21
Given a t-conorm u and an involutive fuzzy complement c, the binary operation i on
[0, 1] defined by
for all is a t-norm such that
〈i, u, c〉is a dual triple.

53. Combinations of operators
Theorem 3.22
Given an involutive fuzzy complement c and an increasing generator of c, the
t-norm and t-conorm generated by are dual with respect to c.

54. Combinations of operators
Theorem 3.23
Let〈i, u, c〉be a dual triple generated by Theorem Then, the fuzzy operations i, u, c satisfy the law of excluded middle and the law of contradiction.

55. Combinations of operators
Theorem 3.24
Let〈i, u, c〉be a dual triple that satisfies the law of excluded middle and the law of contradiction. Then,〈i, u, c〉does not satisfy the distributive laws.

56. Aggregation operations
Axiomatic requirements
Axiom h1.
Axiom h2.

57. Aggregation operations
Axiomatic requirements
Axiom h3.

58. Aggregation operations
Additional requirements
Axiom h4.
Axiom h5.

59. Aggregation operations
Theorem 3.25

60. Aggregation operations
Theorem 3.26

61. Aggregation operations
Theorem 3.27

62. Aggregation operations
Theorem 3.28

63. Exercise 3
3.6
3.7
3.13
3.14