# zbMATH — the first resource for mathematics

Solutions of algebraic equations involving generalized fuzzy numbers. (English) Zbl 0726.65048
A fuzzy number is treated here as a family of intervals $$A=([a_ L(t),a_ R(t)])_{t\in [0,1]}$$ with continuous, piece-wise differentiable functions $$a_ L$$ and $$a_ R$$. Obviously $$a_ L\leq a_ R$$, $$a_ L$$ increases and $$a_ R$$ decreases in [0,1] [cf. R. Goetschel, W. Voxman, Fuzzy Sets Syst. 18, 31-43 (1986; Zbl 0626.26014)]. The paper presents sufficient and necessary conditions for fuzzy numbers A and C such that the families of interval solutions of the equations $$A+X=C,\quad A-X=C,\quad A*X=C,\quad A/X=C$$ form a fuzzy number X.

##### MSC:
 65G30 Interval and finite arithmetic 03E72 Theory of fuzzy sets, etc. 26E50 Fuzzy real analysis
Full Text:
##### References:
  Dubois, D.; Prade, H., Fuzzy real algebra: some results, Fuzzy sets sys., 2, 327-349, (1979) · Zbl 0412.03035  Lukacs, E., Characteristic functions, (1960), Charles Griffin London · Zbl 0201.20404  Mizumoto, M.; Tanaka, K., Some properties of fuzzy numbers, (), 153-164  Moore, R., Interval analysis, (1966), Prentice-Hall Englewood Cliffs, NJ · Zbl 0176.13301  Nguyen, H.T., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004  Sanchez, E., Solution of fuzzy equations with extended operations, Fuzzy sets sys., 12, 237-248, (1984) · Zbl 0556.04001  Thielman, H.P., Theory of functions of real variables, (1953), Prentice-Hall Englewood Cliffs, NJ · Zbl 0051.29201  Yager, R., On solving fuzzy mathematical relationships, Inform. control, 41, 29-55, (1979) · Zbl 0403.03043  Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning, Inform. sci., 8, 199-251, (1975) · Zbl 0397.68071  R. Zhao and R. Govind, Algebraic characteristics of extended fuzzy numbers, Inform. Sci. in press. · Zbl 0774.26015  Zimmermann, H.J., Fuzzy set theory—and its applications, (1985), Kluwer-Nijhoff Boston-Dordrecht-Lancaster · Zbl 0578.90095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.