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

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)

Properties of α-cuts

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

Properties of α-cuts

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

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

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.

Fuzzy set representations

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.

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.

Fuzzy set representations

Extension principle

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

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

Extension principle

Extension principle

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:

Extension principle

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:

Extension principle

Extension principle

Extension principle

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:

Exercise 2 2.4 2.8 2.11