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) = \begin{cases} 1 \text{ if } x \gt 0 \\ 0 \text{ if } x \leq 0 \end{cases}, }[/math]   [math]\displaystyle{ \overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x \gt 0 \\ 1 \text{ if } x \leq 0 \end{cases}. }[/math]

List of implications

No.	Ref.	Year	Implication: {<x, Implication MEMBERSHIP expression, Implication NON-MEMBERSHIP expression >|x ∈ E}

No.	Ref.	Year	Implication MEMBERSHIP expression	Implication NON-MEMBERSHIP expression
→1			max(νA(x),min(μA(x),μB(x)))	min(μA(x),νB(x))
→2			sg(μA(x)-μB(x))	νB(x).sg(μA(x)-μB(x))
→3			1-(1-μ(x)).sg(μA(x)-μB(x))	νB.sg(μA(x)-μB(x))
→4			max(νA(x),μB(x))	min(μA(x),νB(x))
→5			min(1,νA(x)+μB(x))	max(0,μA(x)+νB(x)-1)
→6			νA(x)+μA(x)μB(x)	μA(x)νB(x)
→7			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)))
→8			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))
→9			νA(x)+μA(x)2μB(x)	μA(x)νA(x)+μA(x)2νB(x)
→10			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))
→11			1-(1-μB(x)).sg(μA(x)-μB(x))	νB(x).sg(μA(x)-μB(x)).sg(νB(x)-νA(x))
→12			max(νA(x),μB(x))	1-max(νA(x),μB(x))
→13			νA(x)+μB(x)-νA(x).μB(x)	μA(x).νB(x)
→14			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))
→15			1-sg(μA(x)-μB(x)).sg(νB(x)-νA(x))	sg(sg(μA(x)-μB(x))+sg(νB(x)-νA(x)))
→16			max(sg(μA(x)),μB(x))	min(sg(μA(x)),νB(x))
→17			max(νA(x),μB(x))	min(μA(x).νA(x)+μA(x)2,νB(x))
→18			max(νA(x),μB(x))	min(1-νA(x),νB(x))
→19			max(1-sg(sg(μA(x))+sg(1-νA(x))),μB(x))	min(sg(1-νA(x)),νB(x))
→20			max(sg(μA(x)),sg(μA(x))))	min(sg(μA(x)),sg(μB(x)))
→21			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)))
→22			max(νA(x),1-νB(x))	min(1-νA(x),νB(x))
→23			1-min(sg(1-νA(x)),sg(1-νB(x)))	min(sg(1-νA(x)),sg(1-νB(x)))
→24			sg(μA(x)-μB(x)).sg(νB(x)-νA(x))	sg(μA(x)-μB(x)).sg(νB(x)-νA(x))
→25			max(νA(x),sg(μA(x)).sg(1-νA(x)),μB(x).sg(νB(x)).sg(1-μB(x)))	min(μA(x),νB(x))
→26			max(sg(1-νA(x)),μB(x))	min(sg(μA(x)),νB(x))
→27			max(sg(1-νA(x)),sg(μB(x)))	min(sg(μA(x)),sg(1-νB(x)))
→28			max(sg(1-νA(x)),μB(x))	min(μA(x),νB(x))
→29			max(sg(1-νA(x)),sg(1-μB(x)))	min(μA(x),sg(1-νB(x)))
→30			max(1-μA(x),min(μA(x),1-νB(x)))	min(μA(x),νB(x))
→31			sg(μA(x)+νB(x)-1)	νB(x).sg(μA(x)+νB(x)-1)
→32			1-νB(x).sg(μA(x)+νB(x)-1)	νB(x).sg(μA(x)+νB(x)-1)
→33			1-min(μA(x),νB(x))	min(μA(x),νB(x))
→34			min(1,2-μA(x)-μB(x))	max(0,μA(x)+νB(x)-1)
→35			1-μA(x).νB(x)	μA(x).νB(x)
→36			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)))
→37			1-max(μA(x),νB(x)).sg(μA(x)+νB(x)-1)	max(μA(x),νB(x)).sg(μA(x)+νB(x)-1)
→38			1-μA(x)+(μA(x)2.(1-νB(x)))	μA(x)(1-μA(x))+μA(x)2.νB(x)
→39			(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))
→40			1-sg(μA(x)+νB(x)-1)	1-sg(μA(x)+νB(x)-1)
→41			max(sg(μA(x)),1-νB(x))	min(sg(μA(x)),νB(x))
→42			max(sg(μA(x)),sg(1-νB(x)))	min(sg(μA(x)),sg(1-νB(x)))
→43			max(sg(μA(x)),1-νB(x))	min(sg(μA(x)),νB(x))
→44			max(sg(μA(x)),1-νB(x))	min(μA(x),νB(x))
→45			max(sg(μA(x)),sg(νB(x)))	min(μA(x),sg(1-νB(x)))
→46			max(νA(x),min(1-νA(x),μB(x)))	1-max(νA(x),μB(x))
→47			sg(1-νA(x)-μB(x))	(1-μB(x)).sg(1-νA(x)-μB(x))
→48			1-(1-μB(x)).sg(1-νA(x)-μB(x))	(1-μB(x)).sg(1-νA(x)-μB(x))
→49			min(1,νA(x)+μB(x))	max(0,1-νA(x)-μB(x))
→50			νA(x)+μB(x)-νA(x).μB(x)	1-νA(x)-μB(x)+νA(x).μB(x)
→51			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)))
→52			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))
→53			νA(x)+(1-νA(x))2.μB(x)	(1-νA(x)).νA(x)+(1-νA(x))2.(1-μB(x))
→54			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))
→55			1-sg(1-νA(x)-μB(x))	1-sg(1-νA(x)-μB(x))
→56			max(sg(1-νA(x)),μB(x))	min(sg(1-νA(x)),1-μB(x))
→57			max(sg(1-νA(x)),sg(μB(x)))	min(sg(1-νA(x)),sg(μB(x)))
→58			max(sg(1-νA(x)),sg(1-μB(x)))	1-max(νA(x),μB(x))
→59			max(sg(1-νA(x)),μB(x))	1-max(νA(x),μB(x))
→60			max(sg(1-νA(x)),sg(1-μB(x)))	min(1-νA(x),sg(μB(x)))
→61			max(μB(x),min(νB(x),νA(x)))	min(νB(x),μA(x))
→62			sg(νB(x)-νA(x))	μA(x).sg(νB(x)-νA(x))
→63			1-(1-νA(x)).sg(νB(x)-νA(x))	μA(x).sg(νB(x)-νA(x))
→64			μB(x)+νB(x).νA(x)	νB(x).μA(x)
→65			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))
→66			μB(x)+νB(x)2νA(x)	νB(x).μB(x)+νB(x)2μA(x)
→67			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))
→68			1-(1-νA(x)).sg(νB(x)-νA(x))	μA(x).sg(νB(x)-νA(x)).sg(μA(x)-μB(x))
→69			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))
→70			max(sg((νB(x)),νA(x))	min(sg(νB(x)),μA(x))
→71			max(μB(x),νA(x))	min(νB(x).μB(x)+νB(x)2,μA(x))
→72			max(μB(x),νA(x))	min(1-μB(x),μA(x))
→73			max(1-max(sg(νB(x)),sg(1-μB(x))),νA(x))	min(sg(1-μB(x)),μA(x))
→74			max(sg(νB(x)),sg(νA(x)))	min(sg(νB(x)),sg(νA(x)))
→75			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))
→76			max(μB(x),1-μA(x))	min(1-μB(x),μA(x))
→77			1-min(sg(1-μB(x)),sg(1-μA(x)))	min(sg(1-μB(x)),sg(1-μA(x)))
→78			max(sg(1-μB(x)),νA(x))	min(sg(νB(x)),μA(x))
→80			max(sg(1-μB(x)),νA(x))	min(νB(x),μA(x))
→81			max(sg(1-μB(x)),sg(1-νA(x)))	min(νB(x),sg(1-μA(x)))
→82			max(1-νB(x),min(νB(x),1-μA(x)))	min(νB(x),μA(x))
→83			sg(νB(x)+μA(x)-1)	μA(x).sg(νB(x)+μA(x)-1)
→84			1-μA(x).sg(νB(x)+μA(x)+1)	μA(x).sg(νB(x)+μA(x)+1)
→85			1-νB(x)+νB(x)2.(1-μA(x))	νB(x).(1-νB(x))+νB(x)2
→86			(1-μA(x)).sg(1-νB(x))+sg(1-νB(x))sg(μA(x)+min(1-νB(x),sg(μA(x))))	μA(x).sg(1-νB(x))+νB(x).sg(1-νB(x)).sg(μA(x))
→87			max(sg(νB(x)),1-μA(x))	min(sg(νB(x)),μA(x))
→88			max(sg(νB(x)),sg(1-μA(x)))	min(sg(νB(x)),sg(1-μA(x)))
→89			max(sg(νB(x)),1-μA(x))	min(νB(x),μA(x))
→90			max(sg(νB(x)),sg(μA(x)))	min(νB(x),sg(1-μA(x)))
→91			max(μB(x),min(1-μB(x),νA(x)))	1-max(μB(x),νA(x))
→92			sg(1-μB(x)-νA(x))	min(1-νA(x),sg(1-μB(x)-νA(x)))
→93			1-min(1-νA(x),sg(1-μB(x)-νA(x)))	min(1-νA(x),sg(1-μB(x)-νA(x)))
→94			μB(x)+(1-μB(x))2.νA(x))	(1-μB(x)).μB(x)+(1-μB(x))2.(1-νA(x))
→95			min(νA(x),sg(μB(x)))+sg(μB(x)).(sg(1-νA(x))+min(μB(x),sg(1-νA(x))))	min(1-νA(x),sg(μB(x)))+min(min(1-μB(x),sg(μB(x))),sg(1-νA(x)))
→96			max(sg(1-μB(x)),νA(x))	min(sg(1-μB(x)),1-νA(x)
→97			max(sg(1-μB(x)),sg(νA(x)))	min(sg(1-μB(x)),sg(νA(x)))
→98			max(sg(1-μB(x)),νA(x))	1-max(μB(x),νA(x))
→99			max(sg(1-μB(x)),sg(1-νA(x)))	min(1-μB(x),sg(νA(x)))
→100			max(min(νA(x),sg(μA(x))),μB(x))	min(min(μA(x),sg(νA(x))),νB(x))
→101			max(min(νA(x),sg(μA(x))),min(μB(x),sg(νB(x))))	min(min(μA(x),sg(νA(x))),min(νB(x),sg(μB(x))))
→102			max(νA(x),min(μB(x),sg(νB(x))))	min(μA(x),min(νB(x),sg(μB(x))))
→103			max(min(1-μA(x),sg(μA(x))),1-νB(x))	min(μA(x),sg(1-μA(x)),νB(x))
→104			max(min(1-μA(x),sg(μA(x))),min(1-νB(x),sg(νB(x))))	min(min(μA(x),sg(1-μA(x))),min(νB(x),sg(1-νB(x))))
→105			max(1-μA(x),min(1-νB(x),sg(νB(x))))	min(μA(x),min(νB(x),sg(1-νB(x))))
→106			max(min(νA(x),sg(1-νA(x))),μB(x))	min(min(1-νA(x),sg(νA(x))),1-μB(x))
→107			max(min(νA(x),sg(1-νA(x))),min(μB(x),sg(1-μB(x))))	min(min(1-νA(x),sg(νA(x))),min(1-μB(x),sg(μB(x))))
→108			max(νA(x),min(μB(x),sg(1-μB(x))))	min(1-νA(x),min(1-μB(x),sg(μB(x))))
→109			νA(x)+min(sg(1-μA(x)),μB(x))	μA(x).νA(x)+min(sg(1-μA(x)),νB(x))
→110			max(νA(x),μB(x))	min(μA(x).νA(x)+sg(1-μA(x)),νB(x))
→111			max(νA(x),μB(x).νB(x)+sg(1-μB(x)))	min(μA(x).νA(x)+sg(1-μA(x)),νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x)))
→112			νA(x)+μB(x)-νA(x).μB(x)	μA(x).νA(x)+sg(1-μA(x)).νB(x)
→113			νA(x)+(μB(x).νB(x)-νA(x).(μB(x).νB(x)+sg(1-μB(x)))	(μA(x).νA(x)+sg(1-μA(x))).(νB(x).(μB(x).νB(x)+sg(1-μB(x)))+sg(1-νB(x)))
→114			1-μA(x)+min(sg(1-μA(x)),1-νB(x))	μA(x).(1-μA(x))+min(sg(1-μA(x)),νB(x))
→115			1-min(μA(x),νB(x))	min(μA(x)(1-μA(x))+sg(1-μA(x)),νB(x))
→116			max(1-μA(x),(1-νB(x)).νB(x)+sg(νB(x)))	min(μA(x).(1-μA(x))+sg(1-μA(x)),νB(x).((1-νB(x)).νB(x)+sg(νB(x)))+sg(1-νB(x)))
→117			1-μA(x)-νB(x)+μA(x).νB(x)	(μA(x).(1-μA(x))+sg(1-μA(x))).νB(x)
→118			(1-μA(x)).sg(νB(x))+μA(x).νB(x).(1-νB(x))	(μA(x)-μA(x)2+sg(1-μA(x))).((1-νB(x)).νB(x)2+sg(1-νB(x)))+sg(1-νB(x))(x))
→119			νA(x)+min(sg(νA(x)),μB(x))	(1-νA(x)).νA(x)+min(sg(νA(x)),1-μB(x))
→120			max(νA(x),μB(x))	min((1-νA(x)).νA(x)+sg(νA(x)),1-μB(x))
→121			max(νA(x),μB(x).(1-μB(x))+sg(1-μB(x)))	min((1-νA(x)).νA(x)+sg(νA(x)),(1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x)))+sg(μB(x)))
→122			νA(x)+μB(x)-νA(x).μB(x)	((1-νA(x)).νA(x)+sg(νA(x))).(1-μB(x))
→123			νA(x)+μB(x).(1-μB(x)-νA(x).(μB(x).(1-μB(x))+sg(1-μB(x)))	((1-νA(x)).νA(x)+sg(νA(x))).(((1-μB(x)).(μB(x).(1-μB(x))+sg(1-μB(x))))+sg(μB(x)))
→124			μB(x)+min(sg(1-νB(x)),νA(x))	νB(x).μB(x)+min(sg(1-νB(x)),μA(x))
→125			max(μB(x),νA(x))	min(νB(x).μB(x)+sg(1-νB(x)),μA(x))
→126			max(μB(x),νA(x).μA(x)+sg(1-νA(x)))	min(νB(x).μB(x)+sg(1-νB(x)),μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x)))
→127			μB(x)+νA(x)-μB(x).νA(x)	(νB(x).μB(x)+sg(1-νB(x))).μA(x)
→128			μB(x)+νA(x).μA(x)-μB(x).(νA(x).μA(x)+sg(1-νA(x)))	(νB(x).μB(x)+sg(1-νB(x))).(μA(x).(νA(x).μA(x)+sg(1-νA(x)))+sg(1-μA(x)))
→129			1-νB(x)+min(sg(1-νB(x)),1-μA(x))	νB(x).(1-νB(x))+min(sg(1-νB(x)),μA(x))
→130			1-min(νB(x),μA(x))	min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x))
→131			max(1-νB(x),(1-μA(x)).μA(x)+sg(μA(x)))	min(νB(x).(1-νB(x))+sg(1-νB(x)),μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x)))
→132			1-μA(x).νB(x)	(νB(x).(1-νB(x))+sg(1-νB(x))).μA(x)
→133			1-νB(x)+(1-μA(x)).μA(x)-(1-νB(x)).((1-μA(x)).μA(x)+sg(μA(x)))	(νB(x).(1-νB(x))+sg(1-νB(x))).(μA(x).((1-μA(x)).μA(x)+sg(μA(x)))+sg(1-μA(x)))
→134			μB(x)+min(sg(μB(x)),νA(x))	(1-μB(x)).μB(x)+min(sg(μB(x)),1-νA(x))
→135			max(μB(x),νA(x))	min((1-μB(x)).μB(x)+sg(μB(x)),1-νA(x))
→136			max(μB(x),νA(x).(1-νA(x))+sg(1-νA(x)))	min((1-μB(x)).μB(x)+sg(μB(x)),(1-νA(x)).(νA(x).(1-νA(x))+sg(1-νA(x)))+sg(νA(x)))
→137			μB(x)+νA(x)-μB(x).νA(x)	((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x))
→138			μB(x)+νA(x).(1-νA(x))-μB(x).	((1-μB(x)).μB(x)+sg(μB(x))).(1-νA(x).(νA(x).(1-νA(x)) + sg(1-νA(x)) + sg(νA(x))
→139			(νA(x) + μB(x))/2	(μA(x) + νB(x))/2
→140			(νA(x) + μB(x) + min(νA(x), μB(x)))/3	(μA(x) + νB(x) + max(μA(x), νB(x)))/3
→141			(νA(x) + μB(x) + max(νA(x), μB(x)))/3	(μA(x) + νB(x) + min(μA(x), νB(x)))/3
→142			(3 - μA(x) - νB(x) - max(μA(x), νB(x)))/3	(μA(x) + νB(x) + max(μA(x), νB(x)))/3
→143			(1 - μA(x) + μb(x) + min(1 - μA(x), μB(x)) )/3	(2 + μA(x) - μB(x) - min(1 - μA(x), μB(x)))/3
→144			(1 + νA(x) - νb(x) + min(νA(x), 1 - νB(x)) )/3	(2 - νA(x) + νB(x) + min(νA(x), 1 - νB(x)))/3
→145			(νA(x) + μB(x) + min(νA(x), μB(x)))/3	(3 - νA(x) - μB(x) - min(νA(x), μB(x)))/3
→146			(3 - μA(x) - νB(x) - min(μA(x), νB(x)))/3	(μA(x) + νB(x) + min(μA(x), νB(x)))/3
→147			(1 - μA(x) + μb(x) + max(1 - μA(x), μB(x)) )/3	(2 + μA(x) - μb(x) - max(1 - μA(x), μB(x)) )/3

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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