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Subtractions over intuitionistic fuzzy sets
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For the various definitions of subtraction of over intuitionistic fuzzy sets, the functions sg and sg 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]
List of intuitionistic fuzzy subtractions of —i type
Alternative separated view
List of intuitionistic fuzzy subtractions of —i′ type
| No. | Ref. | Year | Subtraction |
|---|---|---|---|
| —01′ | {<x, min(μA(x), νB(x)), max(νA(x), μB(x))>|x ∈ E} | ||
| —02′ | {<x, min(μA(x), sg(μB(x))), max(νA(x), sg(μB(x)))>|x ∈ E} | ||
| —03′ | {<x, min(μA(x), νB(x)), max(νA(x), μB(x).νB(x) + μB(x)2)>|x ∈ E} | ||
| —04′ | {<x, min(μA(x), νB(x)), max(νA(x), 1 - νB(x))>|x ∈ E} | ||
| —05′ | {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(1 - νB(x)))>|x ∈ E} | ||
| —06′ | {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(μB(x)))>|x ∈ E} | ||
| —07′ | {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), μB(x))>|x ∈ E} | ||
| —08′ | {<x, min(μA(x), 1 - μB(x)), max(νA(x), μB(x))>|x ∈ E} | ||
| —09′ | {<x, min(μA(x), sg(μB(x))), max(νA(x), μB(x))>|x ∈ E} | ||
| —10′ | {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), 1 - νB(x))>|x ∈ E} | ||
| —11′ | {<x, min(μA(x), sg(νB(x))), max(νA(x), sg(νB(x)))>|x ∈ E} | ||
| —12′ | {<x, min(μA(x), νB(x).(μB(x) + νB(x))), max(νA(x), μB(x).(νB(x)2 + μB(x) + μB(x).νB(x)))>|x ∈ E} | ||
| —13′ | {<x, min(μA(x), sg(1 - μB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E} | ||
| —14′ | {<x, min(μA(x), sg(νB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E} | ||
| —15′ | {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E} | ||
| —16′ | {<x, min(μA(x), sg(μB(x))), max(νA(x), sg(1 - μB(x)))>|x ∈ E} | ||
| —17′ | {<x, min(μA(x), sg(1 - νB(x))), max(νA(x), sg(νB(x)))>|x ∈ E} | ||
| —18′ | {<x, min(μA(x), νB(x), sg(μB(x))), max(νA(x), min(μB(x), sg(νB(x))))>|x ∈ E} | ||
| —19′ | {<x, min(μA(x), νB(x), sg(μB(x))), νA(x)>|x ∈ E} | ||
| —20′ | {<x, min(μA(x), νB(x)), νA(x)>|x ∈ E} | ||
| —21′ | {<x, min(μA(x), 1 - μB(x), sg(μB(x))), max(νA(x), min(μB(x), sg(1 - μB(x))))>|x ∈ E} | ||
| —22′ | {<x, min(μA(x), 1 - μB(x), sg(μB(x))), νA(x)>|x ∈ E} | ||
| —23′ | {<x, min(μA(x), 1 - μB(x)), νA(x)>|x ∈ E} | ||
| —24′ | {<x, min(μA(x), νB(x), sg(1 - νB(x))), max(νA(x), min(1 - νB(x), sg(νB(x))))>|x ∈ E} | ||
| —25′ | {<x, min(μA(x), νB(x), sg(1 - νB(x))), νA(x)>|x ∈ E} | ||
| —26′ | {<x, min(μA(x), νB(x)), max(νA(x), μB(x).νB(x) + sg(1 - μB(x)))>|x ∈ E} | ||
| —27′ | {<x, min(μA(x), 1 - μB(x)), max(νA(x), μB(x).(1 - μB(x)) + sg(1 - μB(x)))>|x ∈ E} | ||
| —28′ | {<x, min(μA(x), νB(x)), max(νA(x), (1 - νB(x)).νB(x) + sg(νB(x)))>|x ∈ E} | ||
| —29′ | {<x, min(μA(x), max(0, μB(x).νB(x) + sg(1 - νB(x)))), max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))>|x ∈ E} | ||
| —30′ | {<x, min(μA(x), μB(x).νB(x), max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x)))>|x ∈ E} | ||
| —31′ | {<x, min(μA(x), (1 - μB(x)).μB(x) + sg(μB(x))), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))>|x ∈ E} | ||
| —32′ | {<x, min(μA(x), (1 - μB(x)).μB(x), max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x)))>|x ∈ E} | ||
| —33′ | {<x, min(μA(x), νB(x).(1 - νB(x)) + sg(1 - νB(x))), max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x)))>|x ∈ E} | ||
| —34′ | {<x, min(μA(x), νB(x).(1 - νB(x))), max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x)))>|x ∈ E} |
Alternative separated view
| No. | Ref. | Year | Subtraction:
{<x, Subtraction MEMBERSHIP expression, Subtraction NON-MEMBERSHIP expression >|x ∈ E} |
|---|
| No. | Ref. | Year | Subtraction MEMBERSHIP expression |
Subtraction NON-MEMBERSHIP expression |
|---|---|---|---|---|
| —01′ | min(μA(x), νB(x)) | max(νA(x), μB(x)) | ||
| —02′ | min(μA(x), sg(μB(x))) | max(νA(x), sg(μB(x))) | ||
| —03′ | min(μA(x), νB(x)) | max(νA(x), μB(x).νB(x) + μB(x)2) | ||
| —04′ | min(μA(x), νB(x)) | max(νA(x), 1 - νB(x)) | ||
| —05′ | min(μA(x), sg(1 - νB(x))) | max(νA(x), sg(1 - νB(x))) | ||
| —06′ | min(μA(x), sg(1 - νB(x))) | max(νA(x), sg(μB(x))) | ||
| —07′ | min(μA(x), sg(1 - νB(x))) | max(νA(x), μB(x)) | ||
| —08′ | min(μA(x), 1 - μB(x)) | max(νA(x), μB(x)) | ||
| —09′ | min(μA(x), sg(μB(x))) | max(νA(x), μB(x)) | ||
| —10′ | min(μA(x), sg(1 - νB(x))) | max(νA(x), 1 - νB(x)) | ||
| —11′ | min(μA(x), sg(νB(x))) | max(νA(x), sg(νB(x))) | ||
| —12′ | min(μA(x), νB(x).(μB(x) + νB(x))) | max(νA(x), μB(x).(νB(x)2 + μB(x) + μB(x).νB(x))) | ||
| —13′ | min(μA(x), sg(1 - μB(x))) | max(νA(x), sg(1 - μB(x))) | ||
| —14′ | min(μA(x), sg(νB(x))) | max(νA(x), sg(1 - μB(x))) | ||
| —15′ | min(μA(x), sg(1 - νB(x))) | max(νA(x), sg(1 - μB(x))) | ||
| —16′ | min(μA(x), sg(μB(x))) | max(νA(x), sg(1 - μB(x))) | ||
| —17′ | min(μA(x), sg(1 - νB(x))) | max(νA(x), sg(νB(x))) | ||
| —18′ | min(μA(x), νB(x), sg(μB(x))) | max(νA(x), min(μB(x), sg(νB(x)))) | ||
| —19′ | min(μA(x), νB(x), sg(μB(x))) | νA(x) | ||
| —20′ | min(μA(x), νB(x)) | νA(x) | ||
| —21′ | min(μA(x), 1 - μB(x), sg(μB(x))) | max(νA(x), min(μB(x), sg(1 - μB(x)))) | ||
| —22′ | min(μA(x), 1 - μB(x), sg(μB(x))) | νA(x) | ||
| —23′ | min(μA(x), 1 - μB(x)) | νA(x) | ||
| —24′ | min(μA(x), νB(x), sg(1 - νB(x))) | max(νA(x), min(1 - νB(x), sg(νB(x)))) | ||
| —25′ | min(μA(x), νB(x), sg(1 - νB(x))) | νA(x) | ||
| —26′ | min(μA(x), νB(x)) | max(νA(x), μB(x).νB(x) + sg(1 - μB(x))) | ||
| —27′ | min(μA(x), 1 - μB(x)) | max(νA(x), μB(x).(1 - μB(x)) + sg(1 - μB(x))) | ||
| —28′ | min(μA(x), νB(x)) | max(νA(x), (1 - νB(x)).νB(x) + sg(νB(x))) | ||
| —29′ | min(μA(x), max(0, μB(x).νB(x) + sg(1 - νB(x)))) | max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x))) | ||
| —30′ | min(μA(x), μB(x).νB(x) | max(νA(x), μB(x).(μB(x).νB(x) + sg(1 - νB(x))) + sg(1 - μB(x))) | ||
| —31′ | min(μA(x), (1 - μB(x)).μB(x) + sg(μB(x))) | max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x))) | ||
| —32′ | min(μA(x), (1 - μB(x)).μB(x) | max(νA(x), μB(x).((1 - μB(x)).μB(x) + sg(μB(x))) + sg(1 - μB(x))) | ||
| —33′ | min(μA(x), νB(x).(1 - νB(x)) + sg(1 - νB(x))) | max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x))) | ||
| —34′ | min(μA(x), νB(x).(1 - νB(x))) | max(νA(x), (1 - νB(x)).(νB(x).(1 - νB(x)) + sg(1 - νB(x))) + sg(νB(x))) |
List of intuitionistic fuzzy subtractions of —i′′ type
Alternative separated view
Approaches to defining intuitionistic fuzzy subtractions
References
- Remark on operation "subtraction" over intuitionistic fuzzy sets, Krassimir Atanassov, 2009
- On intuitionistic fuzzy subtraction, generated by an implication from Kleene-Dienes type, Lilija Atanassova, 2009
- On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬11, Beloslav Riečan, Diana Boyadzhieva, Krassimir Atanassov, 2009
- On intuitionistic fuzzy subtraction, related to intuitionistic fuzzy negation ¬4, Beloslav Riečan, Magdaléna Renčová, Krassimir Atanassov, 2009
- Equalities with intuitionistic fuzzy subtractions and negations, Krassimir Atanassov, Magdaléna Renčová, Dimitar Dimitrov, 2010
- On Łukasiewicz's intuitionistic fuzzy subtraction, Beloslav Riečan, Krassimir Atanassov, 2011
- On Zadeh's intuitionistic fuzzy subtraction, Beloslav Riečan, Krassimir Atanassov, 2011
- "What Links Here" References
