By Sebastia Massanet, Joan Torrens (auth.), Anna Maria Fanelli, Witold Pedrycz, Alfredo Petrosino (eds.)

This ebook constitutes the refereed lawsuits of the ninth overseas Workshop on Fuzzy good judgment and functions, WILF 2011 held in Trani, Italy in August 2011.

The 34 revised complete papers offered have been conscientiously reviewed and chosen from 50 submissions. The papers are equipped in topical sections on advances in thought of fuzzy units, advances in fuzzy structures, advances in class and clustering; and applications.

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Extra info for Fuzzy Logic and Applications: 9th International Workshop, WILF 2011, Trani, Italy, August 29-31,2011. Proceedings

Example text

In fact, let us notice that any strict negation N fulfills it (N (x) > 0 for every x < 1). t. the intersection of sets, by using just the first part of Proposition 1 and the left continuity of the t-norm. Proposition 3. Let A be an IF-set on X, let α ∈ (0, 1] and let T be a left continuous t-norm. Then (A)T,N,α = ∩β<α (A)T,N,β . This property enables us to formulate a kind of a representation theorem for IF-sets (see [9]). The equivalence of the next proposition is proved using the results of Proposition 3, while the necessary part involves the construction of a particular IF set A such that μA (x) = sup{γ ∈ (0, 1]|x ∈ Bγ } that it will be showed to satisfy the desired property.

Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Heidelberg (1999) 3. : On the geometrical interpretations of the intuitionistic fuzzy sets. , Szmidt, E. ) Issues in the Representation and Processing of Uncertain and Imprecise Information. Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets, and Related Topics. EXIT, Warsaw (2005) 4. : An algorithm for calculating the threshold of an image representing uncertainty through A-IFSs. In: IPMU 2006, pp. 2383– 2390 (2006) 5. : Image thresholding using intuitionistic fuzzy sets.

N2 : If x ≥ y then N (x) ≤ N (y), ∀x, y ∈ I. In addition, fuzzy negations satisfying the involutive property are called strong fuzzy negations (SFN in short),see [15] and [6]: N3 : N (N (x)) = x, ∀x ∈ U . NS (x) = 1 − x, called the standard negation, is an involutive function on U . An interval function N : U −→ U is an interval fuzzy negation if, for any X, Y in U, the following properties hold: N1 : N([0, 0]) = [1, 1] and N([1, 1]) = [0, 0]. N2a : If X ≥ Y then N(X) ≤ N(Y ); and N2b: If X ⊆ Y then N(X) ⊇ N(Y ).

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