# Generator of Fuzzy Implications
## 2 Theory--New Fuzzy Implication Methods
### The New Proposed Family of Fuzzy Implication
#### 2.3.1 Real Data and Area of Study  
In order to apply the fuzzy implications created, we considered temperature measurements with the corresponding humidity values [49]. The data taken refer to the city of Kavala in Greece and for a period of four months--August, September, October and November of the year 2021--and were taken at the same time, 11:50 a.m. daily [50], for a total of 122 observations. Variable \(x\) represents the humidity values and variable \(y\) represents the temperature values. The values of humidity are between 29% and 94%, while the values of temperature are between 7 \({}^{\circ}\)C and 36 \({}^{\circ}\)C. For the fuzzification of all values, a conversion has been performed on all the values by constructing different fuzzy numbers. 3.2 Implementation of First Step of Methodology Using Matlab Program: The Fuzzification of Real Variables Using Four Membership Degree Functions (Four Cases)  
The authors calculated the degree of membership of the crisp temperature and humidity values using four membership degree functions (isosceles and scalene triangular and isosceles and random trapezium), which were applied in the new type of fuzzy implications for various values of m. Four different models have been created, which are presented in the picture graphs below. The following images were extracted from the fuzzy environment of the Matlab program. First of all, the authors used the Matlab program and applied the following commands on the 122 temperature and humidity values read from excel, respectively:  
Temperature = xlsread (\"temperature.xls\", 1, \"A1:A122\") (Command 1). Humidity = xlsread (\"humidity.xls\", 1, \"A1:A122\") (Command 2). I. First Case--Isosceles trapezium (trapezium membership function)  
In the first case, the authors used graphs in the form of an isosceles trapezium in a rectangular system of axes with abscissa temperatures or humidities, and the ordinates of the corresponding fuzzy numbers [0, 1]. The vertices of the isosceles trapezoids for temperatures are [7, 19, 24, 36] with graph ordinates [0, 1], while the humidities have the abscissas [0.29, 0.59, 0.64, 0.94] with graph ordinates [0, 1]. The data taken refer to the city of Kavala in Greece and a period of four months--August, September, October and November of the year 2021--and were taken at the same time, 11:50, daily [39] for a total of 122 observations. The value of 7 is the lowest temperature value, and value of 36 is the highest temperature value of all 122 temperature values (all values are derived from 122 observations of temperature). The values 19 and 24 are defined by default in Matlab, as they form the vertices of a trapezium (value 19 is equidistant from 7 and value 24 is equidistant from 36; the sum of 7 and 36 is 43 and the sum of 19 and 24 is 43). Specifically, the authors typed in the fuzzy command to open the membership function environment. The following command outputs three degrees of membership by fuzzing the temperature values ranging between 7 and 36 based on the vertices of the isosceles trapezium [7, 19, 24, 36] (see Figure 1). ISOCELESTRAPZEIMtemperature = trapmf (temperature, [7, 19, 24, 36]) (Command 3). Therefore, temperature values greater than 7 and those close to this value will have membership degrees of approximately 0.1 and 0.2, temperature values of 18 and 25 will have a membership degree of approximately 0.9, temperature values between 19 and 24 will have a membership degree of 1, and finally temperature values less than 36 and those close to this value will have membership degrees of about 0.1 and 0.2. Temperature values 7 and 36 have a membership degree of 0. Command 4 below outputs the membership degrees by fuzzing the humidity values ranging from 0.29 to 0.94, based on the vertices of the isosceles trapezium [0.29, 0.59, 0.64, 0.94] (see Figure 2). Figure 1: The procedure for the fuzzification of temperature values ranging from 7 to 36 based on the vertices of the isosceles trapezium [7; 19; 24; 36].