Implications over intuitionistic fuzzy sets: Difference between revisions

Latest revision as of 09:41, 29 April 2022

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

"What Links Here" References

Implications over intuitionistic fuzzy sets: Difference between revisions

Latest revision as of 09:41, 29 April 2022

List of implications

References

See also

Navigation menu

@@ Line 1: / Line 1: @@
 For the various definitions of implication of over [[intuitionistic fuzzy sets]], the functions sg(''x'') and {{overline|sg}}(''x'') have been used:
-<math>
+<math>\text{sg}(x) = \begin{cases} 1  \text{ if } x > 0 \\
-\text{sg}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x > 0 \\
+\text{ if } x \leq 0
-& \text{if } x \leq 0
+\end{cases},</math>&nbsp;&nbsp;&nbsp;<math>
-\end{array},</math>&nbsp;&nbsp;&nbsp;<math>
+\overline{\text{sg}}(x) = \begin{cases} 0 \text{ if } x > 0 \\
-\overline{\text{sg}}(x) = \left \{ \begin{array}{c c} 1 & \text{if } x < 0 \\
+\text{ if } x \leq 0
-& \text{if } x \geq 0
+\end{cases}.</math>
-\end{array}.</math>
-ρ== List of intuitionistic fuzzy implications ==
+<!-- == List of intuitionistic fuzzy implications ==
 {| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;"
@@ Line 539: / Line 538: @@
 |
 | &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(1-μ<sub>A</sub>(x)))</font>, <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>89</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>90</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>91</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),min(1-μ<sub>B</sub>(x),ν<sub>A</sub>(x)))</font>, <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>92</sub>
+|
+|
+| &#123;&#60;x, <font color=green>{{overline|sg}}(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>, <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>93</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>, <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>94</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.ν<sub>A</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.(1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>95</sub>
+|
+|
+| &#123;&#60;x, <font color=green>min(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+sg(μ<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+min(μ<sub>B</sub>(x),sg(1-ν<sub>A</sub>(x))))</font>, <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+min(min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))),sg(1-ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>96</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>97</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font>, <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>98</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>99</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(1-μ<sub>B</sub>(x),{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>100</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),μ<sub>B</sub>(x))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>101</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>102</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>103</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),1-ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>104</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>105</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(1-μ<sub>A</sub>(x),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font>, <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>106</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font>, <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>107</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font>, <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>108</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font>, <font color=red>min(1-ν<sub>A</sub>(x),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>109</sub>
+|
+|
+| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>110</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>111</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>112</sub>
+|
+|
+| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)).ν<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>113</sub>
+|
+|
+| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+(μ<sub>B</sub>(x).ν<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font>, <font color=red>(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x))).(ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>114</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font>, <font color=red>μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>115</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>, <font color=red>min(μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>116</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(1-μ<sub>A</sub>(x),(1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))</font>, <font color=red>min(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>117</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x)-ν<sub>B</sub>(x)+μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x))).ν<sub>B</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>118</sub>
+|
+|
+| &#123;&#60;x, <font color=green>(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))+μ<sub>A</sub>(x).ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))</font>, <font color=red>(μ<sub>A</sub>(x)-μ<sub>A</sub>(x)<sup>2</sup>+{{overline|sg}}(1-μ<sub>A</sub>(x))).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)<sup>2</sup>+{{overline|sg}}(1-ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x))</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>119</sub>
+|
+|
+| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font>, <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>120</sub>
+|
+|
+| &#123;&#60;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>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>121</sub>
+|
+|
+| &#123;&#60;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>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>122</sub>
+|
+|
+| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>, <font color=red>((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>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>123</sub>
+|
+|
+| &#123;&#60;x, <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x)-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>((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>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>124</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>125</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>126</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>127</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>128</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>A</sub>(x)-μ<sub>B</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>129</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-ν<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font>, <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>130</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>, <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>131</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(1-ν<sub>B</sub>(x),(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>132</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>, <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>133</sub>
+|
+|
+| &#123;&#60;x, <font color=green>1-ν<sub>B</sub>(x)+(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)-(1-ν<sub>B</sub>(x)).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font>, <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>134</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font>, <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>135</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>, <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>136</sub>
+|
+|
+| &#123;&#60;x, <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font>, <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),(1-ν<sub>A</sub>(x)).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(ν<sub>A</sub>(x)))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>137</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font>, <font color=red>((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x))</font>&#62;&#124;x &#8712; E&#125;
+|- valign="top"
+| →<sub>138</sub>
+|
+|
+| &#123;&#60;x, <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))-μ<sub>B</sub>(x).</font>, <font color=red></font>&#62;&#124;x &#8712; E&#125;
 |}
+-->
+=== List of implications ===
+{| width="100%" class="wikitable" style="font-family:Courier; font-size:120%;"
+|- valign="top"
+! width="5%" | No.
+! width="5%" | Ref.
+! width="5%" | Year
+! width="85%" | Implication:
+&#123;&#60;x, <font color=green>Implication MEMBERSHIP expression</font>, <font color=red>Implication NON-MEMBERSHIP expression</font> &#62;&#124;x &#8712; E&#125;
-=== Alternative separated view ===
+|}
+{| width="100%" class="wikitable sortable" style="font-family:Courier; font-size:120%;"
+|- valign="top"
+! width="5%" | No.
+! width="5%" | Ref.
+! width="5%" | Year
+! width="40%" | Implication MEMBERSHIP expression<br/>
+! width="45%" | Implication NON-MEMBERSHIP expression<br/>
+|- valign="top"
+| →<sub>1</sub>
+|
+|
+| <font color=green> max(ν<sub>A</sub>(x),min(μ<sub>A</sub>(x),μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>2</sub>
+|
+|
+| <font color=green> {{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red> ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>3</sub>
+|
+|
+| <font color=green> 1-(1-μ<sub></sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>ν<sub>B</sub>.sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)) </font>
+|- valign="top"
+| →<sub>4</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>5</sub>
+|
+|
+| <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font> || <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>6</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>A</sub>(x)μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x)ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>7</sub>
+|
+|
+| <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(μ<sub>A</sub>(x),ν<sub>A</sub>(x)), max(μ<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> || <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)), min(μ<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>8</sub>
+|
+|
+| <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)),sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>9</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x)ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>10</sub>
+|
+|
+| <font color=green>μ<sub>A</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font> || <font color=red>ν<sub>B</sub>.{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>11</sub>
+|
+|
+| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>12</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>13</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>14</sub>
+|
+|
+| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))-ν<sub>B</sub>(x).{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>15</sub>
+|
+|
+| <font color=green>1-sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>sg({{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))+{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>16</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>17</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+μ<sub>A</sub>(x)<sup>2</sup>,ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>18</sub>
+|
+|
+| <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>B</sub>(x))</font>
+|- valign="top"
+| →<sub>19</sub>
+|
+|
+| <font color=green>max(1-sg(sg(μ<sub>A</sub>(x))+sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>20</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(μ<sub>A</sub>(x))))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>21</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).(μ<sub>B</sub>(x)+ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).(μ<sub>A</sub>(x)+ν<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x)<sup>2</sup>+ν<sub>B</sub>(x)+μ<sub>B</sub>(x).ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>22</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),1-ν<sub>B</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>23</sub>
+|
+|
+| <font color=green>1-min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>24</sub>
+|
+|
+| <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>25</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>26</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>27</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>28</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>29</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>30</sub>
+|
+|
+| <font color=green>max(1-μ<sub>A</sub>(x),min(μ<sub>A</sub>(x),1-ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>31</sub>
+|
+|
+| <font color=green>{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>32</sub>
+|
+|
+| <font color=green>1-ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>ν<sub>B</sub>(x).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>33</sub>
+|
+|
+| <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>34</sub>
+|
+|
+| <font color=green>min(1,2-μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>max(0,μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>35</sub>
+|
+|
+| <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>36</sub>
+|
+|
+| <font color=green>min(1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),max(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),max(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font> || <font color=red>max(min(μ<sub>A</sub>(x),ν<sub>B</sub>(x)),min(μ<sub>A</sub>(x),1-μ<sub>A</sub>(x)),min(1-ν<sub>B</sub>(x),ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>37</sub>
+|
+|
+| <font color=green>1-max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>max(μ<sub>A</sub>(x),ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>38</sub>
+|
+|
+| <font color=green>1-μ<sub>A</sub>(x)+(μ<sub>A</sub>(x)<sup>2</sup>.(1-ν<sub>B</sub>(x)))</font> || <font color=red>μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x)<sup>2</sup>.ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>39</sub>
+|
+|
+| <font color=green>(1-ν<sub>B</sub>(x)).{{overline|sg}}(1-μ<sub>A</sub>(x))+sg(1-μ<sub>A</sub>(x)).({{overline|sg}}(ν<sub>B</sub>(x))+(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)))</font> || <font color=red>ν<sub>B</sub>(x).{{overline|sg}}(1-μ<sub>A</sub>(x))+μ<sub>A</sub>(x).sg(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>40</sub>
+|
+|
+| <font color=green>1-sg(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font> || <font color=red>1-{{overline|sg}}(μ<sub>A</sub>(x)+ν<sub>B</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>41</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>42</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),sg(1-ν<sub>B</sub>(x)))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>43</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(sg(μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>44</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>45</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(μ<sub>A</sub>(x)),{{overline|sg}}(ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x),{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>46</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),min(1-ν<sub>A</sub>(x),μ<sub>B</sub>(x)))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>47</sub>
+|
+|
+| <font color=green>{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>48</sub>
+|
+|
+| <font color=green>1-(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>49</sub>
+|
+|
+| <font color=green>min(1,ν<sub>A</sub>(x)+μ<sub>B</sub>(x))</font> || <font color=red>max(0,1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>50</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>51</sub>
+|
+|
+| <font color=green>min(max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),max(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),max(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font> || <font color=red>max(1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x)),min(1-ν<sub>A</sub>(x),ν<sub>A</sub>(x)),min(μ<sub>B</sub>(x),1-μ<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>52</sub>
+|
+|
+| <font color=green>1-(1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x))).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>1-min(ν<sub>A</sub>(x),μ<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>53</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.μ<sub>B</sub>(x)</font> || <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+(1-ν<sub>A</sub>(x))<sup>2</sup>.(1-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>54</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x){{overline|sg}}(ν<sub>A</sub>(x))+sg(ν<sub>A</sub>(x)).({{overline|sg}}(1-μ<sub>B</sub>(x))+ν<sub>A</sub>(x).sg(1-μ<sub>B</sub>(x)))</font> || <font color=red>(1-μ<sub>B</sub>(x)).{{overline|sg}}(ν<sub>A</sub>(x))+(1-ν<sub>A</sub>(x)).sg(ν<sub>A</sub>(x)).sg(1-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>55</sub>
+|
+|
+| <font color=green>1-sg(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>1-{{overline|sg}}(1-ν<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>56</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>57</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),sg(μ<sub>B</sub>(x)))</font> || <font color=red>min(sg(1-ν<sub>A</sub>(x)),{{overline|sg}}(μ<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>58</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>59</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>1-max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>60</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-ν<sub>A</sub>(x)),{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>61</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),min(ν<sub>B</sub>(x),ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>62</sub>
+|
+|
+| <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>63</sub>
+|
+|
+| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>64</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>A</sub>(x)</font>
+|- valign="top"
+| →<sub>65</sub>
+|
+|
+| <font color=green>1-(1-min(μ<sub>B</sub>(x),ν<sub>A</sub>(x))).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>max(ν<sub>B</sub>(x),μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>66</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>ν<sub>A</sub>(x)</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>μ<sub>A</sub>(x)</font>
+|- valign="top"
+| →<sub>67</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+μ<sub>B</sub>(x).sg(1-ν<sub>A</sub>(x)))</font> || <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(1-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>68</sub>
+|
+|
+| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>69</sub>
+|
+|
+| <font color=green>1-(1-ν<sub>A</sub>(x)).sg(ν<sub>B</sub>(x)-ν<sub>A</sub>(x))-μ<sub>A</sub>(x).{{overline|sg}}(ν<sub>B</sub>(x)-ν<sub>A</sub>(x)).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).sg(μ<sub>A</sub>(x)-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>70</sub>
+|
+|
+| <font color=green>max({{overline|sg}}((ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>71</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>,μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>72</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>73</sub>
+|
+|
+| <font color=green>max(1-max(sg(ν<sub>B</sub>(x)),sg(1-μ<sub>B</sub>(x))),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>74</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>75</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(ν<sub>A</sub>(x)+μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).(ν<sub>B</sub>(x)+μ<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x)<sup>2</sup>+μ<sub>A</sub>(x))+ν<sub>A</sub>(x).μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>76</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),1-μ<sub>A</sub>(x))</font> || <font color=red>min(1-μ<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>77</sub>
+|
+|
+| <font color=green>1-min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>78</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>80</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>81</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>82</sub>
+|
+|
+| <font color=green>max(1-ν<sub>B</sub>(x),min(ν<sub>B</sub>(x),1-μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>83</sub>
+|
+|
+| <font color=green>{{overline|sg}}(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)-1)</font>
+|- valign="top"
+| →<sub>84</sub>
+|
+|
+| <font color=green>1-μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font> || <font color=red>μ<sub>A</sub>(x).sg(ν<sub>B</sub>(x)+μ<sub>A</sub>(x)+1)</font>
+|- valign="top"
+| →<sub>85</sub>
+|
+|
+| <font color=green>1-ν<sub>B</sub>(x)+ν<sub>B</sub>(x)<sup>2</sup>.(1-μ<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x)<sup>2</sup></font>
+|- valign="top"
+| →<sub>86</sub>
+|
+|
+| <font color=green>(1-μ<sub>A</sub>(x)).{{overline|sg}}(1-ν<sub>B</sub>(x))+sg(1-ν<sub>B</sub>(x)){{overline|sg}}(μ<sub>A</sub>(x)+min(1-ν<sub>B</sub>(x),sg(μ<sub>A</sub>(x))))</font> || <font color=red>μ<sub>A</sub>(x).{{overline|sg}}(1-ν<sub>B</sub>(x))+ν<sub>B</sub>(x).sg(1-ν<sub>B</sub>(x)).sg(μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>87</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>88</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),sg(1-μ<sub>A</sub>(x)))</font> || <font color=red>min(sg(ν<sub>B</sub>(x)),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>89</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>90</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(ν<sub>B</sub>(x)),{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x),{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>91</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),min(1-μ<sub>B</sub>(x),ν<sub>A</sub>(x)))</font> || <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>92</sub>
+|
+|
+| <font color=green>{{overline|sg}}(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x))</font> || <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>93</sub>
+|
+|
+| <font color=green>1-min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font> || <font color=red>min(1-ν<sub>A</sub>(x),sg(1-μ<sub>B</sub>(x)-ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>94</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.ν<sub>A</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+(1-μ<sub>B</sub>(x))<sup>2</sup>.(1-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>95</sub>
+|
+|
+| <font color=green>min(ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+sg(μ<sub>B</sub>(x)).({{overline|sg}}(1-ν<sub>A</sub>(x))+min(μ<sub>B</sub>(x),sg(1-ν<sub>A</sub>(x))))</font> || <font color=red>min(1-ν<sub>A</sub>(x),{{overline|sg}}(μ<sub>B</sub>(x)))+min(min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))),sg(1-ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>96</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x)</font>
+|- valign="top"
+| →<sub>97</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),sg(ν<sub>A</sub>(x)))</font> || <font color=red>min(sg(1-μ<sub>B</sub>(x)),{{overline|sg}}(ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>98</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>1-max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>99</sub>
+|
+|
+| <font color=green>max({{overline|sg}}(1-μ<sub>B</sub>(x)),{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(1-μ<sub>B</sub>(x),{{overline|sg}}(ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>100</sub>
+|
+|
+| <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>101</sub>
+|
+|
+| <font color=green>max(min(ν<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
+|- valign="top"
+| →<sub>102</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
+|- valign="top"
+| →<sub>103</sub>
+|
+|
+| <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),1-ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>104</sub>
+|
+|
+| <font color=green>max(min(1-μ<sub>A</sub>(x),sg(μ<sub>A</sub>(x))),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(min(μ<sub>A</sub>(x),sg(1-μ<sub>A</sub>(x))),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>
+|- valign="top"
+| →<sub>105</sub>
+|
+|
+| <font color=green>max(1-μ<sub>A</sub>(x),min(1-ν<sub>B</sub>(x),sg(ν<sub>B</sub>(x))))</font> || <font color=red>min(μ<sub>A</sub>(x),min(ν<sub>B</sub>(x),sg(1-ν<sub>B</sub>(x))))</font>
+|- valign="top"
+| →<sub>106</sub>
+|
+|
+| <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),μ<sub>B</sub>(x))</font> || <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),1-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>107</sub>
+|
+|
+| <font color=green>max(min(ν<sub>A</sub>(x),sg(1-ν<sub>A</sub>(x))),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font> || <font color=red>min(min(1-ν<sub>A</sub>(x),sg(ν<sub>A</sub>(x))),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
+|- valign="top"
+| →<sub>108</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),min(μ<sub>B</sub>(x),sg(1-μ<sub>B</sub>(x))))</font> || <font color=red>min(1-ν<sub>A</sub>(x),min(1-μ<sub>B</sub>(x),sg(μ<sub>B</sub>(x))))</font>
+|- valign="top"
+| →<sub>109</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>110</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>111</sub>
+|
+|
+| <font color=green>max(ν<sub>A</sub>(x),μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>112</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x)).ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>113</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+(μ<sub>B</sub>(x).ν<sub>B</sub>(x)-ν<sub>A</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))</font> || <font color=red>(μ<sub>A</sub>(x).ν<sub>A</sub>(x)+{{overline|sg}}(1-μ<sub>A</sub>(x))).(ν<sub>B</sub>(x).(μ<sub>B</sub>(x).ν<sub>B</sub>(x)+{{overline|sg}}(1-μ<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>114</sub>
+|
+|
+| <font color=green>1-μ<sub>A</sub>(x)+min({{overline|sg}}(1-μ<sub>A</sub>(x)),1-ν<sub>B</sub>(x))</font> || <font color=red>μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+min({{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>115</sub>
+|
+|
+| <font color=green>1-min(μ<sub>A</sub>(x),ν<sub>B</sub>(x))</font> || <font color=red>min(μ<sub>A</sub>(x)(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>116</sub>
+|
+|
+| <font color=green>max(1-μ<sub>A</sub>(x),(1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))</font> || <font color=red>min(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x)),ν<sub>B</sub>(x).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)+{{overline|sg}}(ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x)))</font>
+|- valign="top"
+| →<sub>117</sub>
+|
+|
+| <font color=green>1-μ<sub>A</sub>(x)-ν<sub>B</sub>(x)+μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>(μ<sub>A</sub>(x).(1-μ<sub>A</sub>(x))+{{overline|sg}}(1-μ<sub>A</sub>(x))).ν<sub>B</sub>(x)</font>
+|- valign="top"
+| →<sub>118</sub>
+|
+|
+| <font color=green>(1-μ<sub>A</sub>(x)).sg(ν<sub>B</sub>(x))+μ<sub>A</sub>(x).ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))</font> || <font color=red>(μ<sub>A</sub>(x)-μ<sub>A</sub>(x)<sup>2</sup>+{{overline|sg}}(1-μ<sub>A</sub>(x))).((1-ν<sub>B</sub>(x)).ν<sub>B</sub>(x)<sup>2</sup>+{{overline|sg}}(1-ν<sub>B</sub>(x)))+{{overline|sg}}(1-ν<sub>B</sub>(x))</sub>(x))</font>
+|- valign="top"
+| →<sub>119</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),μ<sub>B</sub>(x))</font> || <font color=red>(1-ν<sub>A</sub>(x)).ν<sub>A</sub>(x)+min({{overline|sg}}(ν<sub>A</sub>(x)),1-μ<sub>B</sub>(x))</font>
+|- valign="top"
+| →<sub>120</sub>
+|
+|
+| <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>
+|- valign="top"
+| →<sub>121</sub>
+|
+|
+| <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>
+|- valign="top"
+| →<sub>122</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x)-ν<sub>A</sub>(x).μ<sub>B</sub>(x)</font> || <font color=red>((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>
+|- valign="top"
+| →<sub>123</sub>
+|
+|
+| <font color=green>ν<sub>A</sub>(x)+μ<sub>B</sub>(x).(1-μ<sub>B</sub>(x)-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>((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>
+|- valign="top"
+| →<sub>124</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).μ<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>125</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>126</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>127</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>
+|- valign="top"
+| →<sub>128</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).μ<sub>A</sub>(x)-μ<sub>B</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>(ν<sub>B</sub>(x).μ<sub>B</sub>(x)+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).(ν<sub>A</sub>(x).μ<sub>A</sub>(x)+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>129</sub>
+|
+|
+| <font color=green>1-ν<sub>B</sub>(x)+min({{overline|sg}}(1-ν<sub>B</sub>(x)),1-μ<sub>A</sub>(x))</font> || <font color=red>ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+min({{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>130</sub>
+|
+|
+| <font color=green>1-min(ν<sub>B</sub>(x),μ<sub>A</sub>(x))</font> || <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>131</sub>
+|
+|
+| <font color=green>max(1-ν<sub>B</sub>(x),(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>min(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x)),μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>132</sub>
+|
+|
+| <font color=green>1-μ<sub>A</sub>(x).ν<sub>B</sub>(x)</font> || <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).μ<sub>A</sub>(x)</font>
+|- valign="top"
+| →<sub>133</sub>
+|
+|
+| <font color=green>1-ν<sub>B</sub>(x)+(1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)-(1-ν<sub>B</sub>(x)).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))</font> || <font color=red>(ν<sub>B</sub>(x).(1-ν<sub>B</sub>(x))+{{overline|sg}}(1-ν<sub>B</sub>(x))).(μ<sub>A</sub>(x).((1-μ<sub>A</sub>(x)).μ<sub>A</sub>(x)+{{overline|sg}}(μ<sub>A</sub>(x)))+{{overline|sg}}(1-μ<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>134</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),ν<sub>A</sub>(x))</font> || <font color=red>(1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+min({{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>135</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x))</font> || <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),1-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>136</sub>
+|
+|
+| <font color=green>max(μ<sub>B</sub>(x),ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))</font> || <font color=red>min((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x)),(1-ν<sub>A</sub>(x)).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))+{{overline|sg}}(1-ν<sub>A</sub>(x)))+{{overline|sg}}(ν<sub>A</sub>(x)))</font>
+|- valign="top"
+| →<sub>137</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x)-μ<sub>B</sub>(x).ν<sub>A</sub>(x)</font> || <font color=red>((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x))</font>
+|- valign="top"
+| →<sub>138</sub>
+|
+|
+| <font color=green>μ<sub>B</sub>(x)+ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x))-μ<sub>B</sub>(x).</font> || <font color=red>
+((1-μ<sub>B</sub>(x)).μ<sub>B</sub>(x)+{{overline|sg}}(μ<sub>B</sub>(x))).(1-ν<sub>A</sub>(x).(ν<sub>A</sub>(x).(1-ν<sub>A</sub>(x)) + {{overline|sg}}(1-ν<sub>A</sub>(x)) + {{overline|sg}}(ν<sub>A</sub>(x))
+|- valign="top"
+| →<sub>139</sub>
+|
+|
+| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x))/2</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x))/2</font>
+|- valign="top"
+| →<sub>140</sub>
+|
+|
+| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>141</sub>
+|
+|
+| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + max(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>142</sub>
+|
+|
+| <font color=green>(3 - μ<sub>A</sub>(x) - ν<sub>B</sub>(x) - max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + max(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>143</sub>
+|
+|
+| <font color=green>(1 - μ<sub>A</sub>(x) + μ<sub>b</sub>(x) + min(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> || <font color=red>(2 + μ<sub>A</sub>(x) - μ<sub>B</sub>(x) - min(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>144</sub>
+|
+|
+| <font color=green>(1 + ν<sub>A</sub>(x) - ν<sub>b</sub>(x) + min(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x)) )/3</font> || <font color=red>(2 - ν<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(ν<sub>A</sub>(x), 1 - ν<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>145</sub>
+|
+|
+| <font color=green>(ν<sub>A</sub>(x) + μ<sub>B</sub>(x) + min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font> || <font color=red>(3 - ν<sub>A</sub>(x) - μ<sub>B</sub>(x) - min(ν<sub>A</sub>(x), μ<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>146</sub>
+|
+|
+| <font color=green>(3 - μ<sub>A</sub>(x) - ν<sub>B</sub>(x) - min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font> || <font color=red>(μ<sub>A</sub>(x) + ν<sub>B</sub>(x) + min(μ<sub>A</sub>(x), ν<sub>B</sub>(x)))/3</font>
+|- valign="top"
+| →<sub>147</sub>
+|
+|
+| <font color=green>(1 - μ<sub>A</sub>(x) + μ<sub>b</sub>(x) + max(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font> || <font color=red>(2 + μ<sub>A</sub>(x) - μ<sub>b</sub>(x) - max(1 - μ<sub>A</sub>(x), μ<sub>B</sub>(x)) )/3</font>
+</font>
+|}
 == References ==
@@ Line 560: / Line 1,759: @@
 * [[Issue:On some two-parametric intuitionistic fuzzy implications|On some two-parametric intuitionistic fuzzy implications]], Piotr Dworniczak, 2011
 * [[Issue:Second Zadeh's intuitionistic fuzzy implication|Second Zadeh's intuitionistic fuzzy implication]], Krassimir Atanassov, 2011
+; "What Links Here" References
+{{Special:WhatLinksHere/{{PAGENAME}}|namespace=102|hidetrans=1|hideredirs=1}}
 == See also ==

Implications over intuitionistic fuzzy sets: Difference between revisions

Latest revision as of 09:41, 29 April 2022

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