Minimum about fuzzy logic
Sources of fuzzy logic

## Number sets

Not always have to be a typical cases of sets, e.g. the set of apples and pears. In technical applications, we're often front of sets that are relevant intervals on the axis. In the above examples, we can for example find several sets (intervals) to evaluate water temperature, bearing temperature or intensity of its vibrations. If the value of the input linguistic variables (eg temperature) belongs to one of the sets (eg increased), it is also true the phrase "temperature is increased." It is easier to talk about sets and membership in them, but directly to the veracity and truthfulness of the functions of input statements - entry terms.

## Multi-valued logic and fuzzy logic

A natural generalization of two-valued logic represents the three-valued logic with values, for example, 0 (meaning false), 0.5 (partially true, maybe, unknown), and 1 (true); there exist also other logics with greater number of truth levels. Logical variable in fuzzy logic takes an infinite number of values from the closed interval [0, 1]; the number of values is limited during program implementation and depends on the method of numeric interpretation of the truth value.

## Fuzzy sets

In fuzzy set theory, each element is assigned a degree of its membership in the fuzzy set (membership function) valued in the closed interval [0, 1]. This function is generally designated by the symbol μ, next to which the name of the set is written in subscript; for example μA represents membership of the element in fuzzy set A, μB represents membership of the element in fuzzy set B, μA represents membership in fuzzy set A, μincreased represents membership of the element in fuzzy set increased, etc.

## Belonging to the fuzzy set

It is common that an element of fuzzy set “partially belongs to the set and partially does not belong to the set” (with membership between 0 and 1). The membership of the element in the set may be regarded as fuzzy. Boundary of a fuzzy set is fuzzy as well – meaning vague, hazy blurred. This is also the origin of the word fuzzy. As contrasted to classical sets, it is possible (and common) for fuzzy sets that one element belongs to two or more fuzzy sets with different membership degree at the same time. Thus, it is possible in fuzzy logic to peacefully reconcile conflicts such as “either I am right, or you are” by saying “we both are partially right”. Similarly to classical sets, the system of set operations is defined also for fuzzy sets: among the fundamental ones are operations of fuzzy intersection, union, and complement, but there are other fuzzy set operations as well. Accordingly there exists a tight relationship between set operations and logical operations.

## Composite systems and fuzzy logic

In technical applications of fuzzy systems we often (nearly always) encounter mixed systems, which have input variables in the form of numeric variables (language variables), and logical variables (input language terms) are defined above those variables. While in the two-valued logic transitions of the logical functions of the adjacent terms are sharp (steps), in fuzzy logic they can be gradual and the logical functions can overlap. For example, water temperature of 35 °C can be assessed as partially pleasant and partially already hot; similarly, water temperature of 37 °C can be assessed as partially hot and still partially pleasant.

Fig. 6.2: Possible behavior of logical functions of the terms

## Fuzzy system for bearing diagnostics

For example, when solving the problem of bearing diagnostics we can generalize the process described in the end of Chapter 5. Instead of two-valued terms we will now work with fuzzy terms that are fuzzy variables and take values from the interval [0, 1]. The logical functions for vibration intensity and temperature can have a shape of fractional function (trapezoids and ramps) and are overlapping for the adjacent terms.

Fig. 6.3: Possible behaviour of logical functions of three terms.
Fig. 6.4: Possible behaviour of logical functions of three terms.

## Output fuzzy terms

The result of the evaluation is usually a group of fuzzy variables – output terms, for example with the meaning of OK, warning1, warning2, alarm, malfunction. However, we can desire the result as a value of a single, continuous (numeric) function – the output language function having the meaning of diagnosis of defective bearing. Its value can be determined from the logical values of the output terms. The logical functions of the said terms can have a shape of fractional function (trapezoids or triangles), possibly rectangles, or narrow pulses (singletons).

Fig. 6.5: Possible behaviour of logical functions of five terms.