fimplication_minimal {agop} R Documentation

## Fuzzy Implications

### Description

Various fuzzy implications Each of these is a fuzzy logic generalization of the classical implication operation.

### Usage

fimplication_minimal(x, y)

fimplication_maximal(x, y)

fimplication_kleene(x, y)

fimplication_lukasiewicz(x, y)

fimplication_reichenbach(x, y)

fimplication_fodor(x, y)

fimplication_goguen(x, y)

fimplication_goedel(x, y)

fimplication_rescher(x, y)

fimplication_weber(x, y)

fimplication_yager(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 I: [0,1]\times [0,1]\to [0,1] is a fuzzy implication if for all x,y,x',y'\in [0,1] it holds: (a) if x\le x', then I(x, y)\ge I(x', y); (b) if y\le y', then I(x, y)\le I(x, y'); (c) I(1, 1)=1; (d) I(0, 0)=1; (e) I(1, 0)=0.

The minimal fuzzy implication is given by I_0(x, y)=1 iff x=0 or y=1, and 0 otherwise.

The maximal fuzzy implication is given by I_1(x, y)=0 iff x=1 and y=0, and 1 otherwise.

The Kleene-Dienes fuzzy implication is given by I_{KD}(x, y)=max(1-x, y).

The Lukasiewicz fuzzy implication is given by I_{L}(x, y)=min(1-x+y, 1).

The Reichenbach fuzzy implication is given by I_{RB}(x, y)=1-x+xy.

The Fodor fuzzy implication is given by I_F(x, y)=1 iff x\le y, and max(1-x, y) otherwise.

The Goguen fuzzy implication is given by I_{GG}(x, y)=1 iff x\le y, and y/x otherwise.

The Goedel fuzzy implication is given by I_{GD}(x, y)=1 iff x\le y, and y otherwise.

The Rescher fuzzy implication is given by I_{RS}(x, y)=1 iff x\le y, and 0 otherwise.

The Weber fuzzy implication is given by I_{W}(x, y)=1 iff x<1, and y otherwise.

The Yager fuzzy implication is given by I_{Y}(x, y)=1 iff x=0 and y=0, and y^x otherwise.

### Value

Numeric vector of the same length as x and y. The ith element of the resulting vector gives the result of calculating I(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: fnegation_yager(), tconorm_minimum(), tnorm_minimum()