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Probabilistic implications and probabilistic S-implications, introduced recently by Grzegorzewski, have attracted attention of researchers engaged in fuzzy implications. Since these two families are based on copulas, they form a kind of bridge between probability theory and fuzzy logic. In this paper we generalize the aforementioned two classes and propose a new approach for constructing fuzzy implications which combines a given a priori fuzzy implication and a semicopula.

Fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is considered. New operators that fulfill additional requirements for the core and support of the fuzzy number are suggested. These supplementary conditions guarantee the proper interpretation of the solution even for very skew input fuzzy numbers. Moreover, the trapezoidal approximation without expected interval invariance is also examined and the relationship between these two approximations is discussed.

Trapezoidal approximation of fuzzy numbers is discussed in relation to data aggregation. An important problem whether it is better to simplify initial data before using an aggregation operator or conversely, to aggregate original fuzzy values and then to simplify the output is addressed.

Fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed. Algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed. It is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number, like its ambiguity, width, its value and weighted expected value.

Fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed. The previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number.

The paper contains few comments and positions the paper on (Atanasov's) intuitionistic fuzzy sets presenting points related to terminology, connections with other mathematical structures and possible interpretations.

The problem of the trapezoidal approximation of fuzzy numbers is discussed. A set of criteria for approximation operators is formulated. These constraints can be used for direct operator derivation. A new nearest trapezoidal approximation operator preserving expected interval is suggested.

New methods for measuring distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets, based on the Hausdorff metric, are suggested. The proposed new distances are straightforward generalizations of the well known Hamming distance, the Euclidean distance and their normalized counterparts.

The problem of the interval approximation of fuzzy numbers is discussed. A new interval approximation operator, which is the best one with respect to a certain measure of distance between fuzzy numbers, is suggested.

A problem of defuzzification of a fuzzy test for testing statistical hypotheses with vague data is considered. Two defuzzification operators are discussed. It is shown that the superposition of a fuzzy test and these operators leads to decision strategies equivalent to the classical crisp tests. Moreover, another source of randomization in hypothesis testing is indicated, and a new ground for the randomized tests application is given. Advice to practitioners – how to defuzzify – is also given.

A definition of fuzzy test for testing statistical hypotheses with vague data is proposed. Then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one-sided or two-sided alternative hypotheses is shown. This fuzzy test, contrary to the classical approach, leads not to the binary decision: to reject or to accept given null hypothesis, but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypothesis, respectively. However, it is a natural generalization of the traditional test, i.e. if the data are precise, not vague, we get a classical statistical test with the binary decision. A measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed.