Updates# Fuzzy sets

In natural language, we often use imprecise and ambiguous terms to describe phenomena. We use terms such as tall height, young man, low temperature, big city, etc. But what does it mean if someone is, for example, tall? Is a man who is 180 cm tall tall? Is 179 cm also enough to be considered tall? What if we are talking about a person who is 178 cm? Is it still high, or is it no longer? Where is the line between tall people and those who are not tall? Attempts to establish such a boundary are always subjective and discretionary. This happens almost every time we try to move from a natural description of the world based on qualitative concepts to a quantitative description. To put it more precisely, this happens when we refer in quantitative description to classical logic, based on two logical values: “true” and “false.”

In bivalent logic, membership in a set is decided in an unambiguous, zero-one manner – a given element (object) either belongs to the set or not. There are no intermediate states. Therefore, membership in the set of tall people on the grounds of classical logic must be resolved unequivocally, regardless of the previously indicated unnaturalness of the distinction between tall and not tall people. This problem can be solved by introducing intermediate degrees of membership of elements and using multivalued logic. In value logic, more than two logical values are allowed, so the logical value of a sentence can be not only true (1) and false (0), but also any real number in the range of [0,1], interpreted as the degree of truthfulness of the given sentence. A consequence of the introduction of intermediate logical values is that the degree of membership in the set can also be determined by a number in the range [0,1].

In the case of the set of tall people, this can look as follows: people with a height of less than 150 cm are not considered tall, so we assign them a degree of 0 membership in the set of tall people, people with a height of more than 180 cm, on the other hand, are considered such, so we assign them a degree of membership equal to 1. On the other hand, people between 150 and 180 centimeters tall can be considered tall to some extent, with a degree of adherence intermediate between 0 and 1. The function describing the degree of membership of a set is called the membership function of a fuzzy set. An example graph of the membership function of the fuzzy set of tall people is shown in Figure 1.

The formal definition of a fuzzy set was first proposed in 1965 by Lofti A. Zadeh, in an article *Fuzzy sets [1]*. A fuzzy set is a classical set of elements with their degrees of membership, which can be written:

It is worth noting that the affiliation function is determined subjectively, its choice depends on the user and thus can be tailored to meet the needs of describing specific situations. Nevertheless, there are some standard membership functions and among them can be pointed out the function of Figure 2):

- triangular,
- trapezoidal,
- gaussian,
- sigmoidal

Fuzzy sets are an excellent tool for describing linguistic variables, that is, variables that take imprecise natural language concepts as values.

On fuzzy sets one can define actions that are analogous to actions on classical sets, that is, common part, sum, complement. In the case of fuzzy sets, we can define such operations in (infinitely) many ways. This is because there are many functions that are equivalent to classical operators on sets. The choice of actions on fuzzy sets allows to build a so-called fuzzy system, which makes it possible to carry out inference in a more flexible way than based on classical logic. Such systems have a number of advantages, among which are worth mentioning:

- Stability – small differences in input data generate small differences in results
- allow the input of variables expressed in natural language
- enable the inclusion of expertise.

[1] Zadeh, L. A. (1965). Zadeh, Fuzzy sets. Inform Control, 8, 338-353.

*This article was written thanks to the funds from the European Union’s co-financing of the Operational Program Intelligent Development 2014-2020, a project implemented under the competition of the National Center for Research and Development: under the “Fast Track” competition for micro, small and medium-sized entrepreneurs – competition for projects from less developed regions under Measure 1.1: R&D projects of enterprises Sub-measure 1.1.1 Industrial research and development work carried out by enterprises. Project title: “Developing software to improve forecast accuracy and inventory optimization from the perspective of customer and supplier collaborating in the supply chain using fuzzy deep neural networks.*

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