1. PART 2 Fuzzy sets vs crisp sets
FUZZY SETS AND
FUZZY LOGIC
Theory and Applications
PART 2 Fuzzy sets vs crisp sets
1. Properties of α-cuts
2. Fuzzy set representations
3. Extension principle

2. Properties of α-cuts Theorem 2.1
Let A, B F(X). Then, the following properties hold for all α, β [0, 1]:
(i)
(ii)
(iii)
(iv)
(v)

3. Properties of α-cuts

4. Properties of α-cuts Theorem 2.2
Let Ai F(X) for all i I, where I is an index set. Then,
(vi)
(vii)

5. Properties of α-cuts

6. Properties of α-cuts Theorem 2.3
Let A, B F(X). Then, for all α [0, 1],
(viii)
(ix)

7. Properties of α-cuts Theorem 2.4
For any A F(X), the following properties hold:
(x)
(xi)

8. Fuzzy set representations
Theorem 2.5 (First Decomposition Theorem)
For every A F(X),
where αA is defined by (2.1), and ∪ denotes the standard fuzzy union.

9. Fuzzy set representations

10. Fuzzy set representations
Theorem 2.6 (Second Decomposition Theorem)
For every A F(X),
where α+A denotes a special fuzzy set defined by
and ∪ denotes the standard fuzzy union.

11. Fuzzy set representations
Theorem 2.7 (Third Decomposition Theorem)
For every A F(X),
where Λ(A) is the level of A, αA is defined by (2.1), and ∪denotes the standard fuzzy union.

12. Fuzzy set representations

13. Extension principle

14. Extension principle Extension principle.
Any given function f : X→Y induces two functions,

15. Extension principle
which are defined by for all A F(X) and for all B F(Y).

16. Extension principle

17. Extension principle

18. Extension principle Theorem 2.8
Let f : X→Y be an arbitrary crisp function. Then, for any Ai F(X) and any Bi F(Y),
i I, the following properties of functions obtained by the extension principle hold:

19. Extension principle

20. Extension principle Theorem 2.9
Let f : X→Y be an arbitrary crisp function.
Then, for any Ai F(X) and all α [0, 1]
the following properties of fuzzified by the
extension principle hold:

21. Extension principle

22. Extension principle

23. Extension principle

24. Extension principle Theorem 2.10
Let f : X→Y be an arbitrary crisp function.
Then, for any Ai F(X), f fuzzified by the
extension principle satisfies the equation:

25. Exercise 2
2.4
2.8
2.11