# Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables

^{*}

^{†}

## Abstract

**:**

## 1. Motivation

## 2. Introduction

## 3. Preliminaries

#### 3.1. Fuzzy Implication

**Definition**

**1.**

**Example**

**1.**

#### 3.2. Fuzzy Negations

**Definition**

**2.**

**Definition**

**3.**

**Example**

**2.**

#### 3.3. Triangular Norms (Conjunctions)

**Definition 4.**

## 4. Main Results

#### 4.1. Construction of the Table of the Empiristic Implication

- Low temperature means High humidity.
- Medium temperature means Medium humidity.
- High temperature means Low humidity.

**Remark**

**2.**

#### 4.2. Generated Parametric Fuzzy Implications

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

#### 4.3. Construction of the Table of the Parametric Fuzzy Implication

- Low temperature means High humidity.
- Medium temperature means Medium humidity.
- High temperature means Low humidity.

#### 4.4. Find the Most Appropriate Parametric Fuzzy Implications

#### 4.4.1. The Control of the Norm of the Empiristic and Three Well Known Implications (see Example 1).

- The squared error of the two implications (Empiristic and Kleen-Dienes) gives the result is 8.4922.
- The squared error of the two implications (Empiristic and Lukasiewicz) gives the result is 9.1654.
- The squared error of the two implications (Empiristic and Reichenbach) gives the result is 8.6690.

#### 4.4.2. The Control of the Norm of the Empiristic and Parametric Implication

#### Description the MATLAB code for the Control between Empiristic and Parametric Implication

**Remark**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Data from the Hellenic Meteorological Service (see Reference [1]).

**Figure 2.**The Membership Function of the Temperature Variable is the Gaussian membership functions $f\left(x;\sigma ,c\right)={e}^{-\frac{{(x-c)}^{2}}{2{\sigma}^{2}}}$, from which the standard deviation, σ, and mean, c are gaussmf[5.122 10.72], gaussmf[4.481 19.21], gaussmf[4.768 25.52], respectively.

**Figure 3.**Membership Function of the Humidity Variable is Gaussian membership functions $f\left(x;\sigma ,c\right)={e}^{-\frac{{(x-c)}^{2}}{2{\sigma}^{2}}}$, from which the standard deviation, σ, and mean, c are gaussmf[7.826 50.09], gaussmf[5.371 64.44], gaussmf[6.775 74.89], respectively.

**Figure 4.**Graph of the strong Fuzzy Negations Ν(x) for four random values of the parameter $\alpha $.

**Figure 6.**Relation between Parametric (5) and Square Error.

Designation | Equation |
---|---|

Minimum | ${T}_{M}(x,y)=min\{x,y\}$ |

Algebraic product | ${T}_{p}(x,y)=x\xb7y$ |

Lukasiewicz | ${T}_{LK}(x,y)=max(x+y-1,0)$ |

Active product | ${T}_{D}(x,y)=$$\left\{\begin{array}{cc}0\hfill & \phantom{\rule{1.em}{0ex}}x,y\in [0,1)\hfill \\ min(x,y)\hfill & \phantom{\rule{1.em}{0ex}}otherwise\hfill \end{array}\right.$ |

Nilpotent mimimum | ${T}_{nM}(x,y)=$$\left\{\begin{array}{cc}0\hfill & \phantom{\rule{1.em}{0ex}}x+y\le 1\hfill \\ min(x,y)\hfill & \phantom{\rule{1.em}{0ex}}otherwise\hfill \end{array}\right.$ |

NaN | 0.5857 | 1 | 1 | 1 | 0.4000 | 0.1564 | 0.3682 | 0.5800 | 0.8382 | 1 | 1 |

1 | 2 | 0 | 1 | 3 | 2 | 4 | 8 | 8 | 13 | 14 | 16 |

0.6020 | 1 | 0 | 1 | 5 | 4 | 6 | 14 | 5 | 9 | 15 | 11 |

0.3967 | 0 | 0 | 0 | 4 | 3 | 8 | 7 | 10 | 8 | 10 | 21 |

1 | 0 | 1 | 3 | 4 | 5 | 10 | 7 | 10 | 13 | 12 | 6 |

0.8567 | 1 | 3 | 6 | 3 | 9 | 7 | 6 | 11 | 9 | 8 | 8 |

0.4033 | 3 | 2 | 10 | 6 | 14 | 4 | 3 | 12 | 4 | 8 | 5 |

0.8220 | 4 | 4 | 9 | 9 | 9 | 12 | 8 | 6 | 5 | 3 | 2 |

1 | 6 | 9 | 13 | 14 | 6 | 7 | 6 | 3 | 5 | 1 | 1 |

1 | 8 | 16 | 10 | 8 | 8 | 9 | 6 | 2 | 4 | 0 | 0 |

1 | 19 | 12 | 11 | 11 | 5 | 3 | 5 | 4 | 1 | 0 | 0 |

1 | 27 | 24 | 7 | 4 | 6 | 1 | 1 | 0 | 0 | 0 | 0 |

6.31 | 1 | 75.87 | 10 |

1.58 | 1 | 76.69 | 10 |

5.05 | 1 | 76.11 | 10 |

5.83 | 1 | 74.92 | 10 |

2.26 | 1 | 75.09 | 10 |

4.55 | 1 | 75.30 | 10 |

6.37 | 1 | 77.58 | 10 |

5.42 | 1 | 77.71 | 10 |

4.87 | 1 | 77.89 | 10 |

5.66 | 1 | 77.25 | 10 |

6.20 | 1 | 75.52 | 10 |

4.40 | 1 | 76.21 | 10 |

6.20 | 1 | 75.93 | 10 |

5.33 | 1 | 76.53 | 10 |

0.0282 | 0 | 0.0141 | 0.0423 | 0.0282 | 0.0563 | 0.1127 | 0.1127 | 0.1831 | 0.1972 | 0.2286 |

0.0141 | 0 | 0.0141 | 0.0704 | 0.0563 | 0.0845 | 0.1972 | 0.0704 | 0.1268 | 0.2113 | 0.1571 |

0 | 0 | 0 | 0.0563 | 0.0423 | 0.1127 | 0.0986 | 0.1408 | 0.1127 | 0.1408 | 0.3000 |

0 | 0.0141 | 0.0423 | 0.0563 | 0.0704 | 0.1408 | 0.0986 | 0.1408 | 0.1831 | 0.1690 | 0.0857 |

0.0141 | 0.0423 | 0.0845 | 0.0423 | 0.1268 | 0.0986 | 0.0845 | 0.1549 | 0.1268 | 0.1127 | 0.1143 |

0.0423 | 0.0282 | 0.1408 | 0.0845 | 0.1972 | 0.0563 | 0.0423 | 0.1690 | 0.0563 | 0.1127 | 0.0714 |

0.0563 | 0.0563 | 0.1268 | 0.1268 | 0.1268 | 0.1690 | 0.1127 | 0.0845 | 0.0704 | 0.0423 | 0.0286 |

0.0845 | 0.1268 | 0.1831 | 0.1972 | 0.0845 | 0.0986 | 0.0845 | 0.0423 | 0.0704 | 0.0141 | 0.0143 |

0.1127 | 0.2254 | 0.1408 | 0.1127 | 0.1127 | 0.1268 | 0.0845 | 0.0282 | 0.0563 | 0 | 0 |

0.2676 | 0.1690 | 0.1549 | 0.1549 | 0.0704 | 0.0423 | 0.0704 | 0.0563 | 0.0141 | 0 | 0 |

0.3803 | 0.3380 | 0.0986 | 0.0563 | 0.0845 | 0.0141 | 0.0141 | 0 | 0 | 0 | 0 |

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Makariadis, S.; Souliotis, G.; Papadopoulos, B. Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables. *Symmetry* **2021**, *13*, 509.
https://doi.org/10.3390/sym13030509

**AMA Style**

Makariadis S, Souliotis G, Papadopoulos B. Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables. *Symmetry*. 2021; 13(3):509.
https://doi.org/10.3390/sym13030509

**Chicago/Turabian Style**

Makariadis, Stefanos, Georgios Souliotis, and Basil Papadopoulos. 2021. "Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables" *Symmetry* 13, no. 3: 509.
https://doi.org/10.3390/sym13030509