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}

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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