tnorm_minimum {agop} R Documentation

## t-norms

### Description

Various t-norms. Each of these is a fuzzy logic generalization of the classical conjunction operation.

### Usage

tnorm_minimum(x, y)

tnorm_product(x, y)

tnorm_lukasiewicz(x, y)

tnorm_drastic(x, y)

tnorm_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 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≤ z, then T(x,y)≤ 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 ≤ 1, and min(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 T(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, tconorm_minimum