Applications of critical point theory to variational by Ambrosetti A

By Ambrosetti A

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Simply, f (A) is the range set of f over the domain set A. 4) if f −1 (z) = ∅ 1 z ∈ f (A) . 6) if f −1 (z) = ∅ 1 z ∈ f ([a, b]) . 7) This means that Zadeh’s Extension Principle is the same as the United Extension on Intervals. Let us introduce the Extension Principle for functions of two variables looking towards operations between fuzzy numbers that we will present in the next section. 2 Let f : X × Y −→ Z be a function and let A and B be fuzzy subsets of X and Y , respectively. 8) where f −1 (z) = {(x, y) : f (x, y) = z}.

3, substituting the membership functions by the respective characteristic functions of the sets A1 , A2 , . . , An . The next example illustrates the application of the Cartesian product. 8 which relates the diagnostics of 5 patients with two symptoms: fever and myalgia. 2 To diagnose a patient the doctor evaluates the symptoms that are specific to each disease. Many diseases can present symptoms like fever and myalgia with different intensities and measures. For example, for flu, the patient with fever and myalgia with intensities that, if represented by fuzzy subsets, must have distinct universal sets.

3 The α-levels of the crisp set A + B with the characteristic function χ(A+B) are given by [A + B]α = A + B for all α ∈ [0, 1]. Remember that the intervals A and B are fuzzy sets of the real line, so that the result of this corollary is an immediate consequence of the characteristic function definition of a classical set. The arithmetic operations for fuzzy numbers may be defined from the extension principle for fuzzy sets in analogous way. Actually, they are particular cases of the extension principle where the functions that must be extended are traditional operations for real numbers.

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