# Generator of Fuzzy Implications
## 2 Theory--New Fuzzy Implication Methods
### The New Proposed Family of Fuzzy Implication  
In this section, the authors have analyzed the theorems and proofs of the new proposed family of fuzzy implications. The following theorems are presented:  
**Theorem 1**. : _If \(f:[0,\ 1]\times[0,\ 1]\rightarrow[0,\ 1]\) is a function of the form_  
\[f(x,y)=n(x)V\hat{y}^{m} \tag{1}\]  
_where \(n(x)=1-x\), [46; 47; 48], the function probor \(xVy=x+y-xy\)[32; 39; 41; 45] has been selected for the application of \(t\)-conorm and \(m\) represents the number of probor repetitions, then_  
\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]  
**Proof of Theorem 1. **  
* For \(m=2\) the authors have: \[yVy=\hat{y}^{2}=y+y-y\cdot y=2y-y^{2}\] (3)
* For \(m=3\) the researchers have: \[yVyVy=\hat{y}^{3}=2y-y^{2}+y-(2y-y^{2})\cdot y=3y-3y^{2}+y^{3}\] (4)
* For \(m=4\) the authors have: \[yVyVyVy=\hat{y}^{4}=3y-3y^{2}+y^{3}+y-(3y-3y^{2}+y^{3})\cdot y=4y-6y^{2}+4y^{ 3}-y^{4}\] (5)
* For \(m=5\) the researchers have: \[yVyVyVyVy=\hat{y}^{5}=4y-6y^{2}+4y^{3}-y^{4}+y-\left(4y-6y^{2}+4y^{3}-y^{4} \right)\cdot y=5y-10y^{2}+10y^{3}-5y^{4}+y^{5}\] (6)
* For \(m=6\) the authors have: \[yVyVyVyVyVy=\hat{y}^{6}=5y-10y^{2}+10y^{3}-5y^{4}+y^{5}+y-\left(5y-10y^{2}+10y ^{3}-5y^{4}+y^{5}\right)\cdot y=6y-15y^{2}\] \[+20y^{3}-15y^{4}+6y^{5}-y^{6}\] (7)  
So, generalizing,  
\[\hat{y}^{m}=\binom{m}{1}y-\binom{m}{2}y^{2}+\binom{m}{3}y^{3}-\binom{m}{4}y^ {4}+\binom{m}{5}y^{5}-\dots\binom{m}{m}y^{m} \tag{8}\]  
If \(m\) is even, then (\(-\)) otherwise, If \(m\) is odd, then (\(+\)). \[n(x)Vy=(1-x)Vy=1-x+y-(1-x)\cdot y=1-x+y-y+x\cdot y=1-x+x\cdot y \tag{9}\]  
\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]0  
\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]1  
\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]2  
\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]3  
Therefore, for the fuzzy implication, authors have derived the formula that follows from Theorem 1:\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]4 \[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]5 \[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]6 \[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]7 \[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]8  
Two cases are distinguished depending on the fact that n can be even or odd. Using the binomial Newton, the relationship obtained is:  
\[f(x,y)=1-\sum_{x=0}^{m}(-1)^{x}\binom{m}{\kappa}xy^{x}=1-x(1-y)^{m} \tag{2}\]9  
\[yVy=\hat{y}^{2}=y+y-y\cdot y=2y-y^{2}\]0  
\[yVy=\hat{y}^{2}=y+y-y\cdot y=2y-y^{2}\]1  
and from the binomial Newton, the following formula is obtained. \[yVy=\hat{y}^{2}=y+y-y\cdot y=2y-y^{2}\]2  
Relations (Equations (14) and (16)) are proven analytically by the method of induction. Next, the researchers prove the formula (Equation (14)). We show the relationship inductively by setting \(m\) as equal to 1. We derive the following: In the first member we have \(n(x)V\hat{y}=(1-x)V\hat{y}=1-x+y-(1-x)y=1-x+y-y+xy\)\(=1-x+xy\).