agop (version 0.2.4)

tconorm_minimum: t-conorms

Description

Various t-conorms. Each of these is a fuzzy logic generalization of the classical alternative operation.

Usage

tconorm_minimum(x, y)
tconorm_product(x, y)
tconorm_lukasiewicz(x, y)
tconorm_drastic(x, y)
tconorm_fodor(x, y)

Value

Numeric vector of the same length as x and y. The ith element of the resulting vector gives the result of calculating S(x[i], y[i]).

Arguments

x

numeric vector with elements in $[0,1]$

y

numeric vector of the same length as x, with elements in $[0,1]$

Details

A function $S:[0,1]\times [0,1]\to [0,1]$ is a t-conorm if for all $x,y,z\in [0,1]$ it holds: (a) $S(x,y)=S(y,x)$; (b) if $y\le z$, then $S(x,y)\le S(x,z)$; (c) $S(x,S(y,z))=S(S(x,y),z)$; (d) $S(x,0)=x$.

The minimum t-conorm is given by $S_M(x,y)=max(x,y)$.

The product t-conorm is given by $S_P(x,y)=x+y-xy$.

The Lukasiewicz t-conorm is given by $S_L(x,y)=min(x+y,1)$.

The drastic t-conorm is given by $S_D(x,y)=1$ iff $x,y\in (0,1]$, and $max(x,y)$ otherwise.

The Fodor t-conorm is given by $S_F(x,y)=1$ iff $x+y\ge 1$, and $max(x,y)$ otherwise.

References

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

See Also

Other fuzzy_logic: fimplication_minimal(), fnegation_yager(), tnorm_minimum()