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Implications over intuitionistic fuzzy sets
For the various definitions of implication of over intuitionistic fuzzy sets, the functions sg(x) and sg(x) have been used:
[math]\displaystyle{ \text{sg}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x \gt 0 \\ 0 & \text{if } x \leq 0 \end{array}, }[/math] [math]\displaystyle{ \overline{\text{sg}}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x \lt 0 \\ 0 & \text{if } x \geq 0 \end{array}. }[/math]
ρ== List of intuitionistic fuzzy implications ==
| No. | Ref. | Year | Implication |
|---|---|---|---|
| →1 | {<x, max(νA(x),min(μA(x),μB(x))), min(μA(x),νB(x))>|x ∈ E} | ||
| →2 | {<x, sg(μA(x)-μB(x)), νB(x).sg(μA(x)-μB(x))>|x ∈ E} | ||
| →3 | {<x, 1-(1-μ(x)).sg(μA(x)-μB(x)), νB.sg(μA(x)-μB(x)) >|x ∈ E} | ||
| →4 | {<x, max(νA(x),μB(x)), min(μA(x),νB(x))>|x ∈ E} | ||
| →5 | {<x, min(1,νA(x)+μB(x)), max(0,μA(x)+νB(x)-1)>|x ∈ E} | ||
| →6 | {<x, νA(x)+μA(x)μB(x), μA(x)νB(x)>|x ∈ E} | ||
| →7 | {<x, min(max(νA(x),μB(x)),max(μA(x),νA(x)), max(μB(x),νB(x))), max(min(μA(x),νB(x)), min(μA(x),νA(x)),min(μB(x),νB(x)))>|x ∈ E} | ||
| →8 | {<x, 1-(1-min(νA(x),μB(x))).sg(μA(x)-μB(x)), max(μA(x),νB(x)).sg(μA(x)-μB(x)),sg(νB(x)-νA(x))>|x ∈ E} | ||
| →9 | {<x, νA(x)+μA(x)2μB(x), μA(x)νA(x)+μA(x)2νB(x)>|x ∈ E} | ||
| →10 | {<x, μA(x).sg(1-μA(x))+sg(1-μA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))), νB.sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(1-μB(x))>|x ∈ E} | ||
| →11 | {<x, 1-(1-μB(x)).sg(μA(x)-μB(x)), νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x))>|x ∈ E} | ||
| →12 | {<x, max(νA(x),μB(x)), 1-max(νA(x),μB(x))>|x ∈ E} | ||
| →13 | {<x, νA(x)+μB(x)-νA(x).μB(x), μA(x).νB(x)>|x ∈ E} | ||
| →14 | {<x, 1-(1-μB(x)).sg(μA(x)-μB(x))-νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x)), νB(x).sg(νB(x)-νA(x))>|x ∈ E} | ||
| →15 | {<x, 1-sg(μA(x)-μB(x)).sg(νB(x)-νA(x)), sg(sg(μA(x)-μB(x))+sg(νB(x)-νA(x)))>|x ∈ E} | ||
| →16 | {<x, max(sg(μA(x)),μB(x)), min(sg(μA(x)),νB(x))>|x ∈ E} | ||
| →17 | {<x, max(νA(x),μB(x)), min(μA(x).νA(x)+μA(x)2,νB(x))>|x ∈ E} | ||
| →18 | {<x, max(νA(x),μB(x)), min(1-νA(x),νB(x))>|x ∈ E} | ||
| →19 | {<x, max(1-sg(sg(μA(x))+sg(1-νA(x))),μB(x)), min(sg(1-νA(x)),νB(x))>|x ∈ E} | ||
| →20 | {<x, max(sg(μA(x)),sg(μA(x)))), min(sg(μA(x)),sg(μB(x)))>|x ∈ E} | ||
| →21 | {<x, max(νA(x),μB(x).(μB(x)+νB(x))), min(μA(x).(μA(x)+νA(x)),νB(x).(μB(x)2+νB(x)+μB(x).νB(x)))>|x ∈ E} | ||
| →22 | {<x, max(νA(x),1-νB(x)), min(1-νA(x),νB(x))>|x ∈ E} | ||
| →23 | {<x, 1-min(sg(1-νA(x)),sg(1-νB(x))), min(sg(1-νA(x)),sg(1-νB(x)))>|x ∈ E} | ||
| →24 | {<x, sg(μA(x)-μB(x)).sg(νB(x)-νA(x)), sg(μA(x)-μB(x)).sg(νB(x)-νA(x))>|x ∈ E} | ||
| →25 | {<x, max(νA(x),sg(μA(x)).sg(1-νA(x)),μB(x).sg(νB(x)).sg(1-μB(x))), min(μA(x),νB(x))>|x ∈ E} | ||
| →26 | {<x, max(sg(1-νA(x)),μB(x)), min(sg(μA(x)),νB(x))>|x ∈ E} | ||
| →27 | {<x, max(sg(1-νA(x)),sg(μB(x))), min(sg(μA(x)),sg(1-νB(x)))>|x ∈ E} | ||
| →28 | {<x, max(sg(1-νA(x)),μB(x)), min(μA(x),νB(x))>|x ∈ E} | ||
| →29 | {<x, max(sg(1-νA(x)),sg(1-μB(x))), min(μA(x),sg(1-νB(x)))>|x ∈ E} | ||
| →30 | {<x, max(1-μA(x),min(μA(x),1-νB(x))), min(μA(x),νB(x))>|x ∈ E} | ||
| →31 | {<x, sg(μA(x)+νB(x)-1), νB(x).sg(μA(x)+νB(x)-1)>|x ∈ E} | ||
| →32 | {<x, 1-νB(x).sg(μA(x)+νB(x)-1), νB(x).sg(μA(x)+νB(x)-1)>|x ∈ E} | ||
| →33 | {<x, 1-min(μA(x),νB(x)), min(μA(x),νB(x))>|x ∈ E} | ||
| →34 | {<x, min(1,2-μA(x)-μB(x)), max(0,μA(x)+νB(x)-1)>|x ∈ E} | ||
| →35 | {<x, 1-μA(x).νB(x), μA(x).νB(x)>|x ∈ E} | ||
| →36 | {<x, min(1-min(μA(x),νB(x)),max(μA(x),1-μA(x)),max(1-νB(x),νB(x))), max(min(μA(x),νB(x)),min(μA(x),1-μA(x)),min(1-νB(x),νB(x)))>|x ∈ E} | ||
| →37 | {<x, 1-max(μA(x),νB(x)).sg(μA(x)+νB(x)-1), max(μA(x),νB(x)).sg(μA(x)+νB(x)-1)>|x ∈ E} | ||
| →38 | {<x, 1-μA(x)+(μA(x)2.(1-νB(x))), μA(x)(1-μA(x))+μA(x)2.νB(x)>|x ∈ E} | ||
| →39 | {<x, (1-νB(x)).sg(1-μA(x))+sg(1-μA(x)).(sg(νB(x))+(1-μA(x)).sg(νB(x))), νB(x).sg(1-μA(x))+μA(x).sg(1-μA(x)).sg(νB(x))>|x ∈ E} | ||
| →40 | {<x, 1-sg(μA(x)+νB(x)-1), 1-sg(μA(x)+νB(x)-1)>|x ∈ E} | ||
| →41 | {<x, max(sg(μA(x)),1-νB(x)), min(sg(μA(x)),νB(x))>|x ∈ E} | ||
| →42 | {<x, max(sg(μA(x)),sg(1-νB(x))), min(sg(μA(x)),sg(1-νB(x)))>|x ∈ E} | ||
| →43 | {<x, max(sg(μA(x)),1-νB(x)), min(sg(μA(x)),νB(x))>|x ∈ E} | ||
| →44 | {<x, max(sg(μA(x)),1-νB(x)), min(μA(x),νB(x))>|x ∈ E} | ||
| →45 | {<x, max(sg(μA(x)),sg(νB(x))), min(μA(x),sg(1-νB(x)))>|x ∈ E} | ||
| →46 | {<x, max(νA(x),min(1-νA(x),μB(x))), 1-max(νA(x),μB(x))>|x ∈ E} | ||
| →47 | {<x, sg(1-νA(x)-μB(x)), (1-μB(x)).sg(1-νA(x)-μB(x))>|x ∈ E} | ||
| →48 | {<x, 1-(1-μB(x)).sg(1-νA(x)-μB(x)), (1-μB(x)).sg(1-νA(x)-μB(x))>|x ∈ E} | ||
| →49 | {<x, min(1,νA(x)+μB(x)), max(0,1-νA(x)-μB(x))>|x ∈ E} |
Alternative separated view
References
- On some properties of intuitionistic fuzzy implications, Michał Baczyński, 2003
- Intuitionistic fuzzy implications and Modus Ponens, Krassimir Atanassov, 2005
- A property of intuitionistic fuzzy implications, Yun Shi and Violeta Tasseva, 2005
- On some intuitionistic fuzzy implications, Krassimir Atanassov, 2006
- On a new intuitionistic fuzzy implication of Gaines-Rescher's type, Beloslav Riečan, Krassimir Atanassov, 2007
- A study on some intuitionistic fuzzy implications, Violeta Tasseva, Desislava Peneva, 2007
- On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type, Lilija Atanassova, 2009
- A new intuitionistic fuzzy implication, Lilija Atanassova, 2009
- Intuitionistic fuzzy implications and axioms for implications, Krassimir Atanassov and Dimitar Dimitrov, 2010
- Four modal forms of intuitionistic fuzzy implication →@ and two related intuitionistic fuzzy negations. Part 1, Lilija Atanassova, 2010
- Some Remarks about L. Atanassova’s Paper “A New Intuitionistic Fuzzy Implication”, Piotr Dworniczak, 2010
- On the basic properties of the negations generated by some parametric intuitionistic fuzzy implications, Piotr Dworniczak, 2011
- On some two-parametric intuitionistic fuzzy implications, Piotr Dworniczak, 2011
- Second Zadeh's intuitionistic fuzzy implication, Krassimir Atanassov, 2011
