Soft Computing and Its Applications, Volume One

Fuzzy Sets, Fuzzy Operators, and Fuzzy Relations 89

Lemma 2.1 In case of fuzzy set the law of excluded middle is not valid.

Example 2.26 Let = 1/2, ∀x∈R, then it is easy to see that

~ ∨ = max{~ , } = max{1 − 1/2, 1/2} = 1/2 ≠ 1.

Lemma 2.2 In case of fuzzy set the law of noncontradiction is not valid.

Example 2.27 Let = 1/2, ∀x∈R, then it is easy to see that

~ ∧ = min{~ , } = min{1 − 1/2, 1/2} = 1/2 ≠ 0.

Denition 2.27 The fuzzy union of two fuzzy set A and B on X, denoted A B, is

specied by a function u:

(2.79)

The fuzzy union function u must satisfy at least the following axiomatic require-

ments, for any

• axiom 1 (boundary conditions): ...