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\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.

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 0.2.4 Index]