fimplication_minimal {agop} | R Documentation |

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

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)

`x` |
numeric vector with elements in |

`y` |
numeric vector of the same length as |

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≤ x'*, then *I(x, y)≥ I(x', y)*;
(b) if *y≤ y'*, then *I(x, y)≤ 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≤ y*, and *max(1-x, y)* otherwise.

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

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

The Rescher fuzzy implication is given by *I_{RS}(x, y)=1*
iff *x≤ 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.

Numeric vector of the same length as `x`

and `y`

.
The `i`

th element of the resulting vector gives the result
of calculating `I(x[i], y[i])`

.

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`

[Package *agop* version 0.2-3 Index]