Various t-norms. Each of these is a fuzzy logic generalization of the classical conjunction operation.
tnorm_minimum(x, y)tnorm_product(x, y)
tnorm_lukasiewicz(x, y)
tnorm_drastic(x, y)
tnorm_fodor(x, y)
Numeric vector of the same length as x and y.
The ith element of the resulting vector gives the result
of calculating T(x[i], y[i]).
numeric vector with elements in \([0,1]\)
numeric vector of the same length as x,
with elements in \([0,1]\)
A function \(T: [0,1]\times [0,1]\to [0,1]\) is a t-norm if for all \(x,y,z\in [0,1]\) it holds: (a) \(T(x,y)=T(y,x)\); (b) if \(y\le z\), then \(T(x,y)\le T(x,z)\); (c) \(T(x,T(y,z))=T(T(x,y),z)\); (d) \(T(x, 1)=x\).
The minimum t-norm is given by \(T_M(x,y)=min(x, y)\).
The product t-norm is given by \(T_P(x,y)=xy\).
The Lukasiewicz t-norm is given by \(T_L(x,y)=max(x+y-1,0)\).
The drastic t-norm is given by \(T_D(x,y)=0\) iff \(x,y\in [0,1)\), and \(min(x, y)\) otherwise.
The Fodor t-norm is given by \(T_F(x,y)=0\) iff \(x+y \le 1\), and \(min(x, y)\) otherwise.
Klir G.J, Yuan B., Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall PTR, New Jersey, 1995.
Gagolewski M., Data Fusion: Theory, Methods, and Applications, Institute of Computer Science, Polish Academy of Sciences, 2015, 290 pp. isbn:978-83-63159-20-7
Other fuzzy_logic:
fimplication_minimal(),
fnegation_yager(),
tconorm_minimum()