Please check our Instructions to Authors and send your manuscripts to nifs.journal@gmail.com. Next issue: September/October 2024.
Deadline for submissions: 16 November 2024.
Implications over intuitionistic fuzzy sets: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
Line 731: | Line 731: | ||
| | | | ||
| {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>>|x ∈ E} | | {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>>|x ∈ E} | ||
|- valign="top" | |||
| →<sub>121</sub> | |||
| | |||
| | |||
| {<x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min((1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+{{overline|sg}}(ν<sub>A</sub>(x)),(1-μ<sub>B</sub>(x)).(μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x))+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(μ<sub>B</sub>(x)))</font>>|x ∈ E} | |||
|} | |} | ||
Revision as of 11:53, 14 November 2013
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} | ||
→86 | {<x, (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))>|x ∈ E} | ||
→87 | {<x, max(sg(νB(x)),1-μA(x)), min(sg(νB(x)),μA(x))>|x ∈ E} | ||
→88 | {<x, max(sg(νB(x)),sg(1-μA(x))), min(sg(νB(x)),sg(1-μA(x)))>|x ∈ E} | ||
→89 | {<x, max(sg(νB(x)),1-μA(x)), min(νB(x),μA(x))>|x ∈ E} | ||
→90 | {<x, max(sg(νB(x)),sg(μA(x))), min(νB(x),sg(1-μA(x)))>|x ∈ E} | ||
→91 | {<x, max(μB(x),min(1-μB(x),νA(x))), 1-max(μB(x),νA(x))>|x ∈ E} | ||
→92 | {<x, sg(1-μB(x)-νA(x)), min(1-νA(x),sg(1-μB(x)-νA(x)))>|x ∈ E} | ||
→93 | {<x, 1-min(1-νA(x),sg(1-μB(x)-νA(x))), min(1-νA(x),sg(1-μB(x)-νA(x)))>|x ∈ E} | ||
→94 | {<x, μB(x)+(1-μB(x))2.νA(x)), (1-μB(x)).μB(x)+(1-μB(x))2.(1-νA(x))>|x ∈ E} | ||
→95 | {<x, 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)))>|x ∈ E} | ||
→96 | {<x, max(sg(1-μB(x)),νA(x)), min(sg(1-μB(x)),1-νA(x)>|x ∈ E} | ||
→97 | {<x, max(sg(1-μB(x)),sg(νA(x))), min(sg(1-μB(x)),sg(νA(x)))>|x ∈ E} | ||
→98 | {<x, max(sg(1-μB(x)),νA(x)), 1-max(μB(x),νA(x))>|x ∈ E} | ||
→99 | {<x, max(sg(1-μB(x)),sg(1-νA(x))), min(1-μB(x),sg(νA(x)))>|x ∈ E} | ||
→100 | {<x, max(min(νA(x),sg(μA(x))),μB(x)), min(min(μA(x),sg(νA(x))),νB(x))>|x ∈ E} | ||
→101 | {<x, 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))))>|x ∈ E} | ||
→102 | {<x, max(νA(x),min(μB(x),sg(νB(x)))), min(μA(x),min(νB(x),sg(μB(x))))>|x ∈ E} | ||
→103 | {<x, max(min(1-μA(x),sg(μA(x))),1-νB(x)), min(μA(x),sg(1-μA(x)),νB(x))>|x ∈ E} | ||
→104 | {<x, 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))))>|x ∈ E} | ||
→105 | {<x, max(1-μA(x),min(1-νB(x),sg(νB(x)))), min(μA(x),min(νB(x),sg(1-νB(x))))>|x ∈ E} | ||
→106 | {<x, max(min(νA(x),sg(1-νA(x))),μB(x)), min(min(1-νA(x),sg(νA(x))),1-μB(x))>|x ∈ E} | ||
→107 | {<x, 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))))>|x ∈ E} | ||
→108 | {<x, max(νA(x),min(μB(x),sg(1-μB(x)))), min(1-νA(x),min(1-μB(x),sg(μB(x))))>|x ∈ E} | ||
→109 | {<x, νA(x)+min(sg(1-μA(x)),μB(x)), μA(x).νA(x)+min(sg(1-μA(x)),νB(x))>|x ∈ E} | ||
→110 | {<x, max(νA(x),μB(x)), min(μA(x).νA(x)+sg(1-μA(x)),νB(x))>|x ∈ E} | ||
→111 | {<x, 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)))>|x ∈ E} | ||
→112 | {<x, νA(x)+μB(x)-νA(x).μB(x), μA(x).νA(x)+sg(1-μA(x)).νB(x)>|x ∈ E} | ||
→113 | {<x, 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)))>|x ∈ E} | ||
→114 | {<x, 1-μA(x)+min(sg(1-μA(x)),1-νB(x)), μA(x).(1-μA(x))+min(sg(1-μA(x)),νB(x))>|x ∈ E} | ||
→115 | {<x, 1-min(μA(x),νB(x)), min(μA(x)(1-μA(x))+sg(1-μA(x)),νB(x))>|x ∈ E} | ||
→116 | {<x, 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)))>|x ∈ E} | ||
→117 | {<x, 1-μA(x)-νB(x)+μA(x).νB(x), (μA(x).(1-μA(x))+sg(1-μA(x))).νB(x)>|x ∈ E} | ||
→118 | {<x, (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))>|x ∈ E} | ||
→119 | {<x, νA(x)+min(sg(νA(x)),μB(x)), (1-νA(x)).νA(x)+min(sg(νA(x)),1-μB(x))>|x ∈ E} | ||
→120 | {<x, max(νA(x),μB(x)), min((1-νA(x)).νA(x)+sg(νA(x)),1-μB(x))>|x ∈ E} | ||
→121 | {<x, 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)))>|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