Sketching the Graph of Fuzzy Riemann Integral Based on alpha-Level Sets

Behrouz Fathi-Vajargah, Akram Heidary- Harzavily


In this research, we first study two kinds of fuzzy Riemann integrals. One is based on the crisp compact interval and the other one is considered on the fuzzy interval, where call them as fuzzy Riemann integral of type-I and type-II, respectively. Figure of fuzzy Riemann integral and its sketching based on  level sets to analysis of meaning of fuzzy Riemann integral are proposed in this paper. The Riemann integral and Lebesgue integral are identical for a bounded function on a compact interval, we can apply the Lebesgue's monotone convergence theorem and dominated convergence theorem to prove  that the level set of the fuzzy Riemann integral is a closed interval whose end points are the classical Riemann integrals. We sketch and discuss on the graph of this integral and its concepts based onlevel sets and we finally show that whenever α be closer to one, the length of fuzzy Riemann integral (type-I and type-II) is shorter and near to classic Riemann integral.


fuzzy Riemann integral; level set;

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