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3.15. Plot the 3D and 3D characteristics of the ordinal sum of t-norms where t1= product, t2 = Lukasiewicz and connective, t3 = drastic product.
3.16. Show that the two functions given below

can be used as approximations of the minimum and maximum operation, respectively. Experiment with several values of p to illustrate the quality of this approximation.
3.17. Sometimes membership functions are modeled with the use of B-splines. B-splines are defined recursively. We define the i-th normalized B-spline in the form
- if k = 1

- if k > 1

where i = 0, 1,
, m-k and x0 < x1<
< xm are knots of the spline.
- (i) Plot several low order splines (up to k = 3 or 4).
- (ii) Explain why B-splines can be viewed as fuzzy sets. What would be the main disadvantages, if any, of this approach in modeling membership functions?
3.18. Obstacles can be represented as sets (more precisely, relations), Fig. 3.25. Discuss a usefulness of perceiving the obstacles as fuzzy relations in the problem of obstacle avoidance realized by a small autonomous robot ROBBIE.
Figure 3.25 Obstacles and their fuzzy set representation
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