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}
→50			{<x, νA(x)+μB(x)-νA(x).μB(x), 1-νA(x)-μB(x)+νA(x).μB(x)>|x ∈ E}
→51			{<x, min(max(νA(x),μB(x)),max(1-νA(x),νA(x)),max(μB(x),1-μB(x))), max(1-max(νA(x),μB(x)),min(1-νA(x),νA(x)),min(μB(x),1-μB(x)))>|x ∈ E}
→52			{<x, 1-(1-min(νA(x),μB(x))).sg(1-νA(x)-μB(x)), 1-min(νA(x),μB(x)).sg(1-νA(x)-μB(x))>|x ∈ E}
→53			{<x, νA(x)+(1-νA(x))2.μB(x), (1-νA(x)).νA(x)+(1-νA(x))2.(1-μB(x))>|x ∈ E}
→54			{<x, μB(x)sg(νA(x))+sg(νA(x)).(sg(1-μB(x))+νA(x).sg(1-μB(x))), (1-μB(x)).sg(νA(x))+(1-νA(x)).sg(νA(x)).sg(1-μB(x))>|x ∈ E}
→55			{<x, 1-sg(1-νA(x)-μB(x)), 1-sg(1-νA(x)-μB(x))>|x ∈ E}
→56			{<x, max(sg(1-νA(x)),μB(x)), min(sg(1-νA(x)),1-μB(x))>|x ∈ E}
→57			{<x, max(sg(1-νA(x)),sg(μB(x))), min(sg(1-νA(x)),sg(μB(x)))>|x ∈ E}
→58			{<x, max(sg(1-νA(x)),sg(1-μB(x))), 1-max(νA(x),μB(x))>|x ∈ E}
→59			{<x, max(sg(1-νA(x)),μB(x)), 1-max(νA(x),μB(x))>|x ∈ E}
→60			{<x, max(sg(1-νA(x)),sg(1-μB(x))), min(1-νA(x),sg(μB(x)))>|x ∈ E}
→61			{<x, max(μB(x),min(νB(x),νA(x))), min(νB(x),μA(x))>|x ∈ E}
→62			{<x, sg(νB(x)-νA(x)), μA(x).sg(νB(x)-νA(x))>|x ∈ E}
→63			{<x, 1-(1-νA(x)).sg(νB(x)-νA(x)), μA(x).sg(νB(x)-νA(x))>|x ∈ E}
→64			{<x, μB(x)+νB(x).νA(x), νB(x).μA(x)>|x ∈ E}
→65			{<x, 1-(1-min(μB(x),νA(x))).sg(νB(x)-νA(x)), max(νB(x),μA(x)).sg(νB(x)-νA(x)).sg(μA(x)-μB(x))>|x ∈ E}
→66			{<x, μB(x)+νB(x)2νA(x), νB(x).μB(x)+νB(x)2μA(x)>|x ∈ E}
→67			{<x, νA(x).sg(1-νB(x))+sg(1-νB(x)).(sg(1-νA(x))+μB(x).sg(1-νA(x))), μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(1-νA(x))>|x ∈ E}
→68			{<x, 1-(1-νA(x)).sg(νB(x)-νA(x)), μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x))>|x ∈ E}
→69			{<x, 1-(1-νA(x)).sg(νB(x)-νA(x))-μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x)), μA(x).sg(μA(x)-μB(x))>|x ∈ E}
→70			{<x, max(sg((νB(x)),νA(x)), min(sg(νB(x)),μA(x))>|x ∈ E}
→71			{<x, max(μB(x),νA(x)), min(νB(x).μB(x)+νB(x)2,μA(x))>|x ∈ E}
→72			{<x, max(μB(x),νA(x)), min(1-μB(x),μA(x))>|x ∈ E}
→73			{<x, max(1-max(sg(νB(x)),sg(1-μB(x))),νA(x)), min(sg(1-μB(x)),μA(x))>|x ∈ E}
→74			{<x, max(sg(νB(x)),sg(νA(x))), min(sg(νB(x)),sg(νA(x)))>|x ∈ E}
→75			{<x, max(μB(x),νA(x).(νA(x)+μA(x))), min(νB(x).(νB(x)+μB(x)),μA(x).(νA(x)2+μA(x))+νA(x).μA(x))>|x ∈ E}
→76			{<x, max(μB(x),1-μA(x)), min(1-μB(x),μA(x))>|x ∈ E}
→77			{<x, 1-min(sg(1-μB(x)),sg(1-μA(x))), min(sg(1-μB(x)),sg(1-μA(x)))>|x ∈ E}
→78			{<x, max(sg(1-μB(x)),νA(x)), min(sg(νB(x)),μA(x))>|x ∈ E}
→80			{<x, max(sg(1-μB(x)),νA(x)), min(νB(x),μA(x))>|x ∈ E}
→81			{<x, max(sg(1-μB(x)),sg(1-νA(x))), min(νB(x),sg(1-μA(x)))>|x ∈ E}
→82			{<x, max(1-νB(x),min(νB(x),1-μA(x))), min(νB(x),μA(x))>|x ∈ E}
→83			{<x, sg(νB(x)+μA(x)-1), μA(x).sg(νB(x)+μA(x)-1)>|x ∈ E}
→84			{<x, 1-μA(x).sg(νB(x)+μA(x)+1), μA(x).sg(νB(x)+μA(x)+1)>|x ∈ E}
→85			{<x, 1-νB(x)+νB(x)2.(1-μA(x)), νB(x).(1-νB(x))+νB(x)2>|x ∈ E}

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

See also

• Negations over intuitionistic fuzzy sets
• Subtractions over intuitionistic fuzzy sets
• Cartesian product over intuitionistic fuzzy sets

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