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http://ifigenia.org/wiki/issue:nifs/05/3/31-34
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| Title of paper:
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Measures of contradiction for intuitionistic fuzzy sets and for fuzzy classifications
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| Author(s):
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| Jiri Georg Sustal
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| Brandenburg Technical University of Cottbus, Postfach 10 13 44, D-03013 Cottbus, Germany
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| sustal@math.tu-cottbus.de
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| Presented at:
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Third International Conference on IFSs, Sofia, 16-17 October 1999
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| Published in:
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"Notes on IFS", Volume 5 (1999) Number 3, pages 31—34
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| Download:
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 PDF (184 Kb, Info)
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| Abstract:
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Let us denote by [math]\displaystyle{ u = (u_1,...,u_n) }[/math] a vector of membership degrees (compatibility degrees) of a subject or an element [math]\displaystyle{ x }[/math] from the finite set [math]\displaystyle{ X }[/math] to one of [math]\displaystyle{ n }[/math] classes. Let it hold [math]\displaystyle{ 0 \leq u_i \leq 1 }[/math] for all [math]\displaystyle{ i }[/math], another restriction such as [math]\displaystyle{ \sum_i u_i = 1 }[/math] are not given. In this case the vector [math]\displaystyle{ x }[/math] contains inherently not only the uncertainty about the possible final crisp classification of the element [math]\displaystyle{ x }[/math] into one of [math]\displaystyle{ n }[/math] classes but also, to some extent, a contradiction among its components. Both categories, uncertainty and contradiction, should be treated separately. In the articles, properties of the measures of contradiction are studied and formulas for the evaluation of the contradiction for a given vector [math]\displaystyle{ u }[/math] are proposed. The case [math]\displaystyle{ n = 2 }[/math] is closely related to intuitionistic fuzzy sets and to the intuitionistic fuzzy logic.
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| Keywords:
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Fuzzy partition, intuitionistic fuzzy sets, measures of contradiction, intuitionistic fuzzy logic, many valued logic
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| References:
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- Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, (1986) 87-96.
- Atanassov, K., Gargov, G.: Elements of intuitionistic fuzzy logic. Part 1. Fuzzy Sets and Systems 95, (1998) 39-52.
- Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, 1981.
- Burillo, Р., Bustince, Н., Entropy on intuitionistic fuzzy sets and 011 interval-valued fuzzy sets. Fuzzy Sets a11d Systems 78, (1996) 305-316.
- Sustal, J .G.: On the uncertainty of fuzzy classificatio11s. In: Gupta M.M., Sa11chez Е. (eds.): Approximate Reasoning and Decision Analysis. North-Holland (1982) 125-129.
- Sustal, J.G.: On measures of cluster validity. In: Albrycht J., Wisniewski H.(eds.): Proceedings of the Polish Symposium оп Interval and Fuzzy Mathematics, Poznan (1985), 209-211.
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