tconorm_minimum {agop} R Documentation

## 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)


### 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≤ z, then S(x,y)≤ 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 ≥ 1, and max(x, y) otherwise.

### 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]).

### 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

Other fuzzy_logic: fimplication_minimal, fnegation_yager, tnorm_minimum