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where A11, A12, A21, A22, B1 and B2 are given fuzzy numbers. 4.6. The signal is described as follows where Ω and Φ are Gaussian fuzzy numbers with the membership functions Determine X(t). Plot its membership function for selected time instants (t). Study the same problem assuming that Ω and Φ are intervals defined as follows Compare the results of computing using fuzzy numbers and intervals. 4.7. Consider a fuzzy model governed by the expression where A and B are triangular fuzzy numbers with the membership functions A =(0.5, 1, 2) and B = (0.9, 1, 1.1).
4.8. Consider an addition of two images (fuzzy relations A and B), see Fig. 4.34. Propose pertinent computational formulas. Illustrate your findings for Gaussian membership functions.
4.9. Consider an iterative process of subtracting a triangular fuzzy number A = (1, 0.8, 1.2) from a numeric value B = {10}, Analyze the results of successive iterations and discuss an effect of accumulation of fuzziness. Repeat the same analysis for an iterative multiplication of A and B. 4.10. The classic quadratic map is described as Assuming that A is a triangular fuzzy number, A = (0.5, 0.45, 0.55), analyze the results of iteration of A through this nonlinear mapping. 4.11. Consider the sup-product composition used in the extension principle and perform the addition of two triangular fuzzy numbers. What is the result of this operation? Comment whether you still could retain a triangular fuzzy number as the outcome of this operation. This generalized version of addition of fuzzy numbers has been intensively studied by Dubois and Prade (1981). 4.12. Show that if A1 and A2 are two fuzzy numbers with triangular membership functions with 1/2 overlap and where a1 and a2 are real numbers, then the above expression treated as a function of x, y = y(x) is a piecewise linear function. 4.13. Generalize a triangular membership function to a case of many variables. Assuming the standard way of computing an activation (matching) level of this linguistic label, express your finding as a certain distance function. 4.14. Linguistic quantifiers are fuzzy sets defined in [0, 1] They act on fuzzy sets by transforming their membership functions in the following way Discuss the role of the quantifiers: more or less (τ(v) = v0.5), very (τ(v) = v2) and unknown (τ(v) = 1.0) in transforming fuzzy sets. 4.15. A small robot is equipped with infrared sensors that are used to sense eventual obstacles. Based on the signals of these sensors, the robot can navigate throughout the environment by using two motors that allow it to turn right, left or move forward. Propose some navigation rules. Could you think of some defaults in this navigation protocol? 4.9. References
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