1- Fuzzy logic:

  Fuzzy logic (or fuzzy logic) is a branch of mathematics and computer science that extends the   principles of classical logic to better represent imprecision and uncertainty. Unlike classical logic,   where a proposition is either true (1) or false (0), fuzzy logic allows intermediate values between 0 and   1.

  It was introduced in 1965 by Lotfi A. Zadeh to address complex problems.

2- Linguistique variable: 

A linguistic variable is the triplet that links a variable, its range of values, and the set of fuzzy subsets.

2- Fuzzification:

  Example: Transport company- Yassir-

  Obtained from a customer rating database.

3- Fuzzy operators:

   In logic, there are three main operators: negation (NOT), intersection (AND), and union (OR). These     operators differ from the classical set theory presented by Zadeh and the fuzzy set theory. The         

  following table demonstrates the difference:

4- Defuzzification :

   We assume that :

                            - The driver noted by [1,2,3] is « Bad   »

                            - The driver noted by [4,5] is « Good »

        -      Year > 2018 OR 50 < speed < 60 OR Time > 5 min then note [ 2 , 3 , 4 ]

        -        Year > 2003 OR 50 < speed < 60 OR Time > 5 min then note [ 2 , 3 , 4 ]

    4- Characteristics of a fuzzy set:

   Definition 1: The height of Y, denoted h(Y), is the upper bound of the arrival function of its belonging     function: h(Y) = sup {AY(x)|x∈X}

   Definition 2: The support of Y is any element x of X from which supp(Y)={x∈X|AY(x)>0}

   Definition 3: The nucleus of Y is any element x of X from which supp(Y)={x∈X|AY(x)=h(Y)}

   Definition 4: an α-cut of Y is any element x of X from which α-cut(x)={x X|AY(x) α}

5- Characteristics of a fuzzy set:

6- Fuzzy logic applications:

Automation: Control of machines such as washing machines or air conditioners.

Artificial intelligence: Recommendation systems, image recognition and natural language processing.

Medicine: Medical diagnosis based on blurred symptoms.

Robotics: Navigation systems where precise decisions are difficult to make.

Risk management: Assessing probabilities in uncertain environments.

Fuzzy logic is particularly useful when data is vague, incomplete or uncertain. It offers a flexible and intuitive approach to model complex systems.